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7. INDEFINITE INTEGRALS

: Important Formulae :

(1) ∫ xn dx = x n + 1 + C (2) ∫ ax dx = ax

+ C n + 1 log a

(3) ∫ ex dx = ex

+ C (4) ∫ dx / x = log x + C

(5) ∫ sin x dx = – cos x + C (6) ∫ cos x dx = sin x + C

(7) ∫ tan x dx = log | sec x | + C (8) ∫ cot x dx = log | sin x | + C

(9) ∫ sec x dx = log | sec x + tan x | + C (10) ∫cosec x dx = log |cosec x – cot x| + C = log tan x + + C = – log |cosec x + cot x |+ C 2 4 = log | tan ( x / 2 ) | + C

(11) ∫ sec2x dx = tan x + C (12) ∫ cosec2x dx = – cot x + C

(13) ∫ sec x . tan x dx = sec x + C (14) ∫ cosec x . cot x dx = – cosec x + C

(15) ∫ dx / x √x2 – 1 = sec –1 x + C (16) ∫ –dx / x √x2 – 1 = cosec –1 x + C

: MAGICAL ‘9’ FORMULAE :

(1) ∫ dx / ( x2 + a2 ) = tan–1( x / a ) + C (2) ∫ dx / ( x2 – a2 ) = 1 log x – a + C a 2a x + a

(3) ∫ dx / ( a2 – x2 ) = 1 log a + x + C (4) ∫ dx / √a2 – x2 = sin–1( x / a ) + C 2a a – x

(5) ∫ dx / √a2 + x2 = log x + √a2 + x2 + C (6) ∫dx / √x2 – a2 = log x + √x2 – a2 + C

(7) ∫ √a2 + x2 dx = x √a2 + x2 + a2 log x + √a2 + x2 + C 2 2

(8) ∫ √x2 – a2 dx = x √x2 – a2 – a2 log x + √x2 – a2 + C 2 2

(9) ∫ √a2 – x2 dx = x √a2 – x2 + a2 sin–1( x / a ) + C 2 2


: Rules of Integration :

(1) ∫ k f(x) dx = k . ∫ f(x) dx + C (2) ∫ { f(x) g(x) } dx = ∫ f(x)dx ∫ g(x) dx + C

(3) ∫ d{ f(x) } = f(x) + C : Methods of Integration :

Method of Substitution :

(1) If integral is of the type (i) ∫ f(x). f1(x) dx , (ii) ∫f1(x) dx / f(x) (iii) ∫ [f(x)]n. f1(x) dx

(iv) ∫g{f(x)} f1(x) dx . in each of the above cases we put f(x) = t , then integral reduces to,

(i) ∫ t dt (ii) ∫dt / t (iii) ∫ [t]n. dt (iv) ∫g(t) dt .

(2) If the integral of the type ∫f(ax + b) dx , we put ax + b = t , Differentiating, w.r.t x adx = dt dx = dt / a,

Then integral becomes, ∫ f(t) dt / a , with ax + b = t.

(3) If the integral is of the type, ∫ sinnx.cosmx dx . (i) If ‘n’ is odd put cosx = t (ii) If ‘m’ is odd put sin x = t (iii) If n & m both is odd put sin x = t , n ≥ m cos x = t , m > n (iv) If n & m both is even, no substitution is made. We use the transformation , 2sin2x = 1 – cos2x , 2cos2x = 1 + cos2x

(4) If in integral is of the form ∫dx / { ax2 + bx + c } and ax2 + bx + c = 0 has no real roots then follow the following steps Step – I : make the coefficient of x2 unity by taking ‘a’ common. Step – II : add and subtract the square of half of coefficient ‘x’ i.e, ( b / 2a)2. Step – III : apply the formula applicable, out of

∫ dx / ( x2 + a2 ) , ∫ dx / ( x2 – a2 ) , ∫ dx / ( a2 – x2 )

(5) If in integral is of the form ∫dx / √ax2 + bx + c . Follow the following steps, Step – I : make the coefficient of x2 unity by taking ‘a’ common. Step – II : add and subtract the square of half of coefficient of ‘x’ i.e, ( b / 2a)2. Step – III : apply the formula applicable, out of

∫ dx /√x2 + a2 , ∫ dx / √x2 – a2 , ∫ dx / √a2 – x2


(6) If in integral is of the form ∫√ax2 + bx + c dx . Follow the following steps, Step – I : make the coefficient of x2 unity by taking ‘a’ common. Step – II : add and subtract the square of half of coefficient of ‘x’ i.e, ( b / 2a)2. Step – III : apply the formula applicable, out of

∫√x2 + a2 dx , ∫√x2 – a2 dx , ∫√a2 – x2 dx

(7) If in integral is of the form ∫( px + q )dx / { ax2 + bx + c } and ax2 + bx + c = 0 has no real roots. Then follow the following steps Step – I : put px + q = A d { ax2 + bx + c } + B , find A & B by equating the dx coefficient of like terms. Step – II : The integral will reduces to A {rule 1(ii) } + B {rule(4)}

(8) If in integral is of the form ∫( px + q )dx / √ax2 + bx + c . Then follow the following steps Step – I : put px + q = A d { ax2 + bx + c } + B , find A & B by equating the dx coefficient of like terms. Step – II : The integral will reduces to A {rule 1(iii) } + B {rule(5)}

(9) If in integral is of the form ∫( px + q )√ax2 + bx + c dx. Then follow the following steps

Step – I : put px + q = A d { ax2 + bx + c } + B , find A & B by equating the dx coefficient of like terms. Step – II : The integral will reduces to A {rule 1(iii) } + B {rule(6)}

(10) If integral is of the form ∫dx / {asin2x + bcos2x} ; ∫dx / {a + bcos2x} ; ∫dx / {a + bsin2x} ;

∫dx / {asinx + bcosx}2 ; ∫dx / {asin2x + bcos2x + c }.Then do the following steps Steps – I : Divide the numerator( Nr.) and denominator( Dnr.) by cos2x . Steps – II : Replace sec2x, if any in Dnr. by 1 + tan2x. Steps – III : Put tan x = t , so that the sec2x dx = dt and the integral takes the form

∫dt / { at2 + bt + c }. Now follow the rule (4).

(11) If integral is of the form ∫dx / {a sin x + b cos x} ; ∫dx / {a + b cos x } ; ∫dx / {a + b sin x} ;

∫dx / { a sin x + b cos x + c }. Then do the following steps

Steps – I : Put sin x = 2tan( x / 2) and cos x = 1 – tan2( x / 2 ) 1 + tan2( x / 2 ) 1 + tan2( x / 2 ) Steps – II : Replace the numerator’s 1 + tan2( x / 2 ) = sec2(x / 2 ). Steps – III : Put tan( x / 2 ) = t sec2( x / 2 ) dx = 2 dt. and the integral takes

the form : ∫dt / { at2 + bt + c }. Now follow the rule (4).


(12) If integral is of the form ∫( a sin x + b cos x )dx / { c sin x + d cos x } Then follow the following steps Step – I : put a sin x + b cos x = A d {c sin x + d cos x } + B {c sin x + d cos x } dx find A & B by equating the coefficient of like terms. Step – II : The integral will reduces to A {rule 1(ii) } + B x .

(13) If integral is of the form ∫( p sin x + q cos x + r )dx / { a sin x + b cos x + c } Then follow the following steps Step – I : put p sin x + q cos x + r = A d {a sin x + b cos x + c } dx + B {a sin x + b cos x + c } + C find A , B & C by equating the coefficient of like terms. Step – II : The integral will reduces to A {rule 1(ii) } + B x + C { rule ( 11 )} .

(14) If integral is of the form ∫ √풂풙 + 풃 풅풙√풄풙+ 풅

Then follow the following steps

Step – I : Rationalize the numerator Step – II : Apply rule 8.

(15) If integral is of the form ∫ dx / (ax + b) √px + q . To solve such integral put px + q = t2.

(16) If integral is of the form ∫ dx / (ax2 + bx + c) √px + q . To solve such integral put px + q = t2.

(17) If integral is of the form ∫ dx / (ax + b) √px2 + qx + r . To solve such integral put ax + b = 1 / t.

(18) If integral is of the form ∫ dx / (ax2 + b) √px2 + q . To solve such integral put x = 1 / t.

(19) If integral is of the form ∫dx / { x4 + λx2 + 1 } ; ∫( x2 + 1 )dx / { x4 + λx2 + 1 }

∫( x2 – 1 )dx / { x4 + λx2 + 1} with λ R , Then follow the following steps Steps – I : divide the numerator( Nr.) and denominator( Dnr.)by x2. Steps – II : put x + 1 = t or x – 1 = t , whichever on differentiation gives Nr. x x

Method of Partial Fraction : If integral is of the form ∫ f(x) dx / g(x) Applicability :

(1) If degree of f(x) is less than degree of g(x), Otherwise i.e, degree of f(x) ≥ degree of g(x), do the

direct division first. The integral thus reduces to p(x) + 풒(풙)품 (풙)

with degree of q(x) < degree of g(x)

(2) The denominator g(x) has at least two factor, Otherwise, i.e, g(x) = 0 has no real root, follow the rule (7).

Partial Fractions : If integral ∫ f(x) dx / g(x) holds both of the above two applicability conditions i.e, degree of f(x) < degree of g(x) & g(x) has at least two factors. Then following partial fractions are taken according to the form of f(x).


Form of Rational Function Form of Partial Fraction

풇(풙)풅풆품풓풆풆 ≤ ퟏ(풙 − 풂)(풙 − 풃)

푨(풙 풂) +

푩(풙 풃)

풇(풙)풅풆품풓풆풆 ≤ ퟏ(풙 − 풂)ퟐ

푨(풙 풂) +

푩(풙 풂)ퟐ

풇(풙)풅풆품풓풆풆 ≤ ퟐ(풙 − 풂)(풙 − 풃)(풙 − 풄)

푨(풙 풂) +

푩(풙 풃) +

푪(풙 풄)

풇(풙)풅풆품풓풆풆 ퟐ(풙 풂)ퟐ.(풙 풃)

푨(풙 풂) +

푩(풙 풂)ퟐ +

푪(풙 풃)

풇(풙)풅풆품풓풆풆 ≤ ퟐ(풙 − 풂)ퟑ

푨(풙 풂) +

푩(풙 풂)ퟐ +

푪(풙 풂)ퟑ

풇(풙)풅풆품풓풆풆 ≤ ퟐ(풙 − 풂)(풑풙ퟐ + 풒풙 + 풓)

푨(풙 풂) +

푩풙 푪풑풙ퟐ 풒풙 풓

풇(풙)풅풆품풓풆풆 ≤ ퟑ(풂풙ퟐ + 풃풙 + 풄)(풑풙ퟐ + 풒풙 + 풓)

푨풙 푩풂풙ퟐ 풃풙 풄

+ 푩풙 푪

풑풙ퟐ 풒풙 풓

Finding Values of A, B, C - - -. We find the values of A, B, C - - - by equating the coefficients of the terms having the equal degrees of ‘x’. However by putting the values such as x = 0, ± 1, ± 2, - - -, we get the values of A, B, C - - - , but this process is O.K for objectives only. From CBSE XII point of view and according to NCERT students are advised not to follow the second ( however easier) process, this process will fetch ‘0’ Marks. Method of By Parts : If f(x) and g(x) be any two functions then according to By Parts

∫ f(x).g(x) dx = f(x) . ∫g(x) dx – ∫ { f 1(x) ∫g(x) dx} dx Here, f(x) is the first function and g(x) is the second function. However, the first and second function is determined by ‘ILATE’ rule. I → Inverse Function : sin–1 f(x) , cos–1 f(x), tan–1 f(x) . L → Logarithmic Function : log(f(x)) A → Algebraic Function : polynomials a xn + b xn – 1 + - - - + c T → Trigonometrical Function : sin{f(x)}, cos{f(x)}- - - E → Exponential Function : e f (x) , a f (x) , 2 f (x).

(19) If integral is of the form ∫ ex { f (x) + f 1(x) }dx = ex f (x) + C

(20) If integral is of the form ∫{f (x) g1(x) + f 1(x) g(x)}dx = f (x) . g(x) + C

*********


PROBLEMS ON INDEFINITE INTEGRAL Problems on Substitution :

(1) ∫ sec2x dx / cosec2x (2) ∫dx / (1 + sin x)

(3) ∫sin x dx / sin (x + a ) (4) ∫ ( x3 – 1 )1 / 3. x 5 dx

(5) ∫ (e2x – 1) dx / (e2x + 1) *(6) ∫dx / ( 1 + tan x)

(7) ∫ √tan x dx / sinx . cosx (8) ∫ sin3x . cos2x dx

(9) ∫dx / sin2x . cos2x (10) ∫ x3 sin ( tan–1 x4 ) dx / (1 + x8 )

(11) ∫cos 2x .cos 4x . cos 6x dx (12) ∫ sin3(2x +1) dx

*(13) ∫ cos4 2x dx (14) ∫ sin3x .cos3x dx

(15) ∫ sin2x dx / (1 + cos x) (16) ∫tan3 2x .sec 2x dx

*(17) ∫ tan4x dx *(18) ∫ dx / sinx . cos3x

*(19) ∫dx / cos( x – a).cos(x – b) (20) ∫ex(1 + x ) dx /cos2(ex x)

(21) ∫ (x – 1)dx / √x2 – 1 (22) ∫dx / ( 3x2 + 13x – 10)

(23) ∫ (x + 2)dx / (2x2 + 6x + 5) *(24) ∫ (x + 3)dx / √5 – 4x – x2

(25) ∫dx / √(x – a)(x – b) *(26) ∫ ( 6x + 7)dx / √ (x – 5)(x – 4)

(27) ∫ ( x + 2 )dx / √4x – x2 (28) ∫ ( x + 2 ) dx / √x2 + 2x + 3

*(29) ∫ {√tan x + √cot x } dx *(30) ∫dx / (x1 / 2 + x1 / 3 )

(31) ∫dx / cos (x + a).cos (x + b) (32) ∫dx / √sin3x . sin ( x + a )

(33) ∫sin2x cos2x.dx / √9 – cos4(2x) (34) ∫ ( 5x + 3 ) dx / (x 2 + 4x + 10) .

(35) ∫ (e5 log x – e4 log x ) dx / (e3 log x – e2 log x). (36) ∫ dx / { √x + a + √x + b }


*(37) ∫ √1 – √x / 1 + √x dx *(38) ∫ (x3 – x2 + x – 1) dx / (x – 1)

(39) ∫ tan4√x . sec2√x dx / √x (40) ∫dx / ( 1 – tan x)

*(41) ∫dx / ( 1 + cot x) *(42) ∫ dx / √5x2 – 2x

*(43) ∫ √x2 + 1 { log (x2 + 1) – 2 log x}dx / x4 . (44) ∫dx / {ex + e–x } Problems on Partial Fractions :

*(45) ∫ (3sinx – 2 ) cosx dx / {5 – cos2x – 4sinx} *(46) ∫ (x2 + 1)dx / {x2 – 5x + 6}

*(47) ∫ x2 dx / (x2 + 1)(x2 + 4) (48) ∫ ( 1 – x2 )dx / x(1 – 2x)

*(49) ∫ (x2 + 1)(x2 + 2)dx / (x2 + 3)(x2 + 4) *(50) ∫ (x3 + x + 1)dx / (x2 – 1)

(51) ∫ x dx / (x2 + 1)(x – 1) (52) ∫ 2dx / (1 – x)(1 + x2)

(53) ∫ (3x + 5)dx / {x3 – x2 – x + 1} (54) ∫ 2x dx / (x2 + 1)(x2 + 3)

(55) ∫ dx / (x2 + 1)(x2 + 4) (56) ∫ dx / (ex – 1) Problems on By Parts :

*(57) ∫ (x2 + 1)ex dx / (x + 1)2 *(58) ∫ (x – 3)ex dx / (x – 1)3

(59) ∫ x ex dx / ( 1 + x)2 *(60) ∫ [log(log x) + {1 / log x}2] dx

*(61) ∫ x sin–1x dx *(62) ∫ x tan–1x dx

*(63) ∫ (sin–1x )2 dx (64) ∫ x sec2x dx

(65) ∫ x ( log x )2 dx (66) ∫ e2 x sin x dx

*(67) ∫ sin–1{ 2x / ( 1 + x2 )}dx *(68) ∫ tan–1{√1 – x / √1 + x }dx

*(69) ∫{sin–1√x – cos–1√x}dx / (sin–1√x + cos–1√x) *(70) ∫ (2 + sin 2x )ex dx / (1 + cos 2x)

(71) ∫ √x2 + 3x dx *(72) ∫ √1 + 3x – x2 dx

(73) ∫√x2 – 8x + 7 dx (74) ∫ √1 + (x2 / 9) dx *******


HOTS PROBLEMS Problems on Substitution :

(1) ∫ (sin6x + cos6x) dx / sin2x. cos2x (2) ∫ ( 2x + 3x )dx / 5x.

(3) ∫ tan x . tan 2x . tan 3x dx (4) ∫ tan x dx / ( a + b tan2x)

(5) ∫ dx / sin(x – a).cos(x – b) (6) ∫dx / sin(a – x) . sin(x – b)

(7) ∫ dx / cos(a – x) . cos(x – b) *(8) ∫ √ퟏ + 풙 풅풙√풙

*(9) ∫ √1 – sin x dx *(10) ∫ √sec x – 1 dx

(11) ∫ dx / √1 + sin x (12) ∫ √cosec x – 1 dx

(13) ∫ sin(x – a) dx / sin(x + a) *(14) ∫ √sin(x – a) dx / √sin(x + a)

*(15) ∫√a – x dx / √a + x (16) ∫ sin2x dx / sin3x . sin 5x

(17) ∫ (x – 1) dx / √x + 1 (18) ∫dx / {√x + √x + 1 }

(19) ∫ dx / x . ( xn + 1 ) (20) ∫ (x4 + 1)dx / (x2 + 1)

(21) ∫ sin2x . cos2x dx *(22) ∫ tan–1(sec x + tan x) dx

*(23) ∫ √1 + 2 tan x ( sec x + tan x ) dx (24) ∫ tan–1{sin 2x / (1 + cos 2x) } dx

(25) ∫cot –1{sin 2x / (1 – cos 2x)} dx (26) ∫tan–1{√1 – cos2x / √1 + cos2x}dx

*(27) ∫dx / {sin3 / 4 x . cos5 / 4 x } (28) ∫{ex – 1 + xe – 1}dx / (ex + xe )

(29) ∫ (1 – tan x)dx / (1 + tan x) (30) ∫{e3a log x + e3x log a}dx

*(31) ∫{x2 + 3x + 2}dx / (x – 2)(x + 1) (32) ∫dx / x2(x4 + 1)3 / 4.

*(33) ∫ (x3 + x)dx / (x4 – 9) (34) ∫ sin x dx / sin 3x

*(35) ∫dx / {2sin x + 3cos x}2 *(36) ∫dx / {1 + sin x + cos x}


(37) ∫ (1 + sin x)dx / sin x ( 1 + cos x) (38) ∫x √a2 – x2 dx / √a2 + x2

(39) ∫ dx / (4cos x – 1) (40) ∫dx / (2 + 3 cos x – 4 sinx)

(41) ∫ dx / (3 + 2 sin x + cos x) (42) ∫dx / (4cos x – 7 sin x)

(43) ∫dx / (5 cos x + 7 sin x) (44) ∫sin x dx / ( sin x – cos x)

*(45) ∫ (2sin x + 3cos x)dx / (4sin x + 5cos x) (46) ∫ (5sin x + 6cos x)dx / (2sin x + 3cos x)

(47) ∫ (3cos x + 2)dx / (sin x + 2cos x + 3) (48) ∫ dx / (5 + 7cos x + sin x)

*(49) ∫dx / ( 1 + 3sin2x + 8cos2x) *(50) ∫dx / (1 + 3sin2x)

*(51) ∫dx / (sin x + sec x) *(52) ∫dx / (cos x + cosec x)

(53) ∫dx / (a sin x + b cos x)2 (54) ∫ √1 + sec x dx

*(55) ∫ (x2 – 1)dx / (x2 + 1).√x3 + x2 + x (56) ∫dx / x(x5 + 2)

*(57) ∫ sec4x dx (58) ∫cos x dx / cos(x – a)

(59) ∫dx / (sin2x – 4cos2x) *(60) ∫ √5 – x dx / √x + 2

(61) ∫dx / sin3 / 2x . cos1 / 2x (62) ∫dx / (1 – sin x)

(63) ∫ (x – 1)dx / (x + 1).√x3 + x2 + x *(64) ∫sin2x dx /{a2 sin2x + b2 cos2x}

(65) ∫dx / [x.log x . log(log x) . log{log(log x)}] (66) ∫sinx dx / √1 + sin x

(67) ∫sec x dx / (a + b tan x) (68) ∫ x dx / √x4 + 4

*(69) ∫ (sin x + cos x)dx / √sin2x (70) ∫dx / (x + 1)(x2 + 2x + 2)

(71) ∫dx / (3x + 1)(9x2 + 6x + 6) *(72) ∫dx / √ e2x + 3ex + 1

*(73) ∫ (cos x + sin x)dx / (9 + 16sin 2x) *(74) ∫cos x dx / (cos x + sin x )2

(75) ∫ (cos x – sin x)dx / √4 + 5cos 2x (76) ∫cos7x dx


(77) ∫ sin2x . cos4x dx (78) cos5x dx / sin2x

(79) ∫x dx / {√x2 + a2 + √x2 – a2 } (80) ∫2x dx / √1 – 4x

(81) ∫sec x . cosec x dx / log( cot x) (82) ∫x1 / 3dx / { x1 / 2 – x1 / 3 }

(83) ∫x √1 – x dx / √1 + x (84) ∫dx / (x2 + 1) .√x2 – 1

(85) ∫dx / (x – a) √x2 – a2 (86) ∫ (2 – x2)dx / (1 + x) √1 – x2

(87) ∫ secmx . tan x dx *(88) ∫cos9x dx / sin x

*(89) ∫ (cos x – sin x)dx / sin2x *(90) ∫√cos2x . dx / sin x

(91) ∫cosec x dx / log tan {(π / 2) + (x / 2) } (92) ∫dx / (sec x + cosec x ) (93) If f1(x) = 1 / 2 & f(1) = 5 / 2, then prove that f(x) = x + 2 2 (94) If f1(x) = a sin x + b cos x ; f1(0) = 4, f(0) = 3 and f(π / 2) = 5 , then prove that f(x) = 2 cos x + 4 sin x + 1 .

(95) ∫ dx / (a + b tan x) Problems on By Parts :

*(96) ∫ x2sin–1x dx *(97) ∫ x2 tan–1x dx

(98) ∫ sin( √x ) dx (99) ∫ cos–1( √x ) dx

(100) ∫ tan–1( √x ) dx (101) ∫ sec–1( √x ) dx

*(102) ∫ sin–1{√x / √x + a } dx *(103) ∫sec3x dx

(104) ∫sin(log x)dx (105) ∫ x2. sin2x dx

(106) ∫x . sec2(2x) dx (107) ∫x2. sin x . cos x dx

(108) ∫ x. sin x . sin 2x . sin 3x dx (109) ∫x2 log(1 + x) dx

(110) ∫ (log x)2 dx (111) ∫ex . sin x . cos x . cos 2x dx


(112) ∫ ex(1 + x). log( x.ex )dx (113) ∫xn(log x)2 dx

(114) ∫cos2x . log{(cos x + sin x) / (cos x – sin x )}dx (115) ∫cos–1{(1 – x2) / (1 + x2)}dx

(116) ∫tan–1{(3x – x3) / (1 – 3x2)}dx (117) ∫tan–1{2x / (1 – x2)}dx

(118) ∫ x. cos3x . sinx dx *(119) ∫ (x + sin x)dx / (1 + cos x)

(120) ∫ea x . sin( bx + c)dx (121) ∫x2. sin–1x dx / (1 – x2)3 / 2

*(122) ∫ x2 dx / (x sin x + cos x)2 *(123) ∫ log(1 + x)dx / √1 + x

(124) ∫ex . cos2x dx (125) ∫ (1 + x2) cos 2x dx

(126) ∫x2n – 1.cos( xn ) dx *(127) ∫cos(√x) dx

(128) ∫ e m tan – 1 x dx / (1 + x2) 3 / 2 (129) ∫ log | x + √a2 + x2 |dx

(130) ∫x2.ex dx / (x + 2)2 (131) ∫ (x + 1).ex dx / (x + 2)2

(132) ∫ex( tan x + log sec x )dx (133) ∫ex{(2 – sin2x) / (1 – cos 2x)}dx

(134) ∫ex{sec x + log(sec x + tan x)}dx (135) ∫ logx / (1 + log x )2 dx

*(136) ∫ (sin log x + cos log x ) dx (137) ∫ (tan log x + sec2 log x )dx

(138) ∫ex{(x – 2) / x3 }dx (139)∫ex{log(1 + cos2x) – 2tan x}dx

(140) ∫ e2x{(sin4x – 2) / (1 – cos4x)}dx (141) ∫ex( cot x + log sin x )dx

(142) ∫cosx . √9 – sin2x dx *(143) ∫ √ퟐ풙+ ퟑ dx / √ퟑ풙 − ퟐ

(144) ∫ ex{(1 / √x2 + a2 ) + log | x + √a2 + x2 | }dx *(145) ∫ √x2 + 4x + 1 dx

(146) ∫√7x – 10 – x2 dx (147) ∫ √(x – 1)(2 – x)dx

*(148) ∫x √1 + x – x2 dx *(149) ∫ (x – 5)√x2 + x dx

(150) ∫√(x – 1)(x – 2) dx (151) ∫ (x + 3)√3 – 4x – x2 dx


Problems on Partial Fraction :

(152) ∫ (3x + 2)dx / (x3 – 6x2 + 11x – 6) (153) ∫ (2x – 1)dx / (x + 1)(x2 + 2)

(154) ∫ (x3 – 6x2 + 10x – 2)dx / (x2 – 5x + 6) *(155) ∫ x3 dx / (x – 1)(x – 2)

*(156) ∫ (x2 + 4x + 3)dx / (x2 + x – 2) (157) ∫ (x2 – x – 2)dx / (1 – x2)

*(158) ∫ dx / (x – x3) (159) ∫ dx / (x2 + x)(x2 – 1)

(160) ∫dx / x(1 + x2) *(161) ∫ dx / sin x . cos2x

(162) ∫ (1 – cos x)dx / sin x (1 + cos x) *(163) ∫sec2 x dx / {tan3 x + 4tan x}

(164) ∫ (1 – cos x)dx / cos x (1 + cos x) *(165) ∫dx / (sin x – sin2x)

(166) ∫ cos x . dx / (2 + sinx)(3 + 4sin x) *(167) ∫dx / (1 + 3ex + 2e2x )

(168) ∫ tan x dx / (1 – sin x) (169) ∫ dx / cos x (5 – 4sin x)

*(170) ∫dx / sin x (3 + 2 cos x) (171) ∫ cos2x dx / (cos2x + 4 sin2x)

*(172) ∫dx / (sin3x + cos3x) (173) ∫dx / ( x3 + x4 )

(174) ∫ x4 dx / (x – 1)(x2 + 1) (175) ∫ (x + 1)dx / x4(x – 1)

(176) ∫dx / x(x + 1)3 (177) ∫ (x2 + 2)dx / (x – 1)(x – 2)3

*(178) ∫ (x2 + x + 1)dx / (x + 1)2.(x + 2) (179) ∫dx / (1 + cot 3 x) Problems on Special Integrals :

(180) ∫ (x2 + 1) dx / (x4 + 1) (181) ∫ (x2 – 1) dx / (x4 + 1)

(182) ∫ x2. dx / (x4 + 1) *(183) ∫dx / (x4 + 1)

(184) ∫ (x2 + 1) dx / (x4 + x2 + 1) (185) ∫ (x2 – 1) dx / (x4 + x2 + 1)

(186) ∫ x2. dx / (x4 + x2 + 1) *(187) ∫dx / (x4 + x2 + 1)


(188) ∫ (x2 + 9) dx / (x4 + 81) (189) ∫ ( x2 + 4 ) dx / ( x4 + 16 )

*(190) ∫ dx / { sin4x + cos4x} *(191) ∫ √cot x dx

(192) ∫dx / (2x + 3) √4x + 5 (193) ∫dx / (x + 1) √x + 2

(194) ∫dx / (3x + 2). √5 – x (195) ∫dx / (x2 + 1) √x

(196) ∫dx / (x2 + 3) √x + 4 (197) ∫dx / (x – 1) √x2 + 4

(198) ∫dx / (x2 + 4) √x2 + 1 (199) ∫dx / x2 .√x + 1

(200) ∫√1 + x2 dx / ( 1 – x2 ) (201) ∫ (x + 1) dx / (x – 1) √x + 2

(202) ∫ (x + 2) dx / (x2 + 3x + 3) √x + 1 (203) ∫ √x – 1dx / x . √x + 1

(204) ∫dx / (x – 1). √1 + x + x2 (205) ∫x √1 – x2 dx / √1 + x2

*(206) ∫ sin2x.dx / { sin4x + cos4x} *(207) ∫ sin x . dx / √1 + sin x

(208) ∫dx / { x4 – 5x2 + 16 } (209) ∫(x2 + 1) dx / (x4 + 7x2 + 1)

*(210) ∫ (2 sin2x – cos x ) dx / {6 – cos2x – 4 sin x } . *(211) ∫ √tan x dx

******

ANSWERS PROBLEMS ON INDEFINITE INTEGRAL (1) tan x – x + C (2) tan x – sec x + C (3) x cos a – sin a . log |sin (x + a) | + C

(4) (x3 – 1) 7 / 3 + (x3 – 1) 4 / 3 + C (5) log | e x + e – x | + C (6) x + log |sin x + cos x| + C 7 4 (7) 2 √tan x + C (8) cos5 x – cos3 x + C (9) tan x – cot x + C 5 3 (10) – cos (tan–1 x4 ) + C (11) 1 . { 2 sin 12x + 3 sin 8x + 6 sin 4x }+ C 4 96 (12) 1 .{cos3 (2x + 1) – 3 cos (2x + 1)} + C (13) 1 .{24x + 8 sin 4x + sin 8x} + C 6 64


(14) 2.cos6 x – 3cos4 x + C (15) x – sin x + C (16) sec3 2x – 3 sec 2x + C 12 6 (17) tan3 x – tan x + x + C (18) log | tan x | + tan2 x + C (19) 1 . log cos(x – a) + C 3 2 sin(a – b) cos(x – b)

(20) tan( xex ) + C (21) √x2 – 1 – log | x + √x2 – 1 | + C (22) 1 . log 3x – 2 + C 17 x + 5

(23) 1 . log | 2x2 + 6x + 5 | + 1 . tan–1 (2x + 3) + C (24) –√5 – 4x – x2 + sin–1 x + 3 + C 4 2 3 (25) log x – a + b + √(x – a)(x – b) + C 2 (26) 6 √x2 – 9x + 20 + 34 . log x – 9 + √x2 – 9x + 20 + C (27) – √4x – x2 + 4 sin–1 x – 2 + C 2 2

(28) √x2 + 2x + 3 + log | x + 1 + √x2 + 2x + 3 | + C (29) √2 . sin–1( sin x – cos x ) + C (30) 2.√x – 3 x 1/ 3 + 6 x 1 / 6 – 6 log ( 1 + x 1 / 6 ) + C (31) 1 . log cos (x + b) + C sin (a – b) cos (x + a)

(32) – 2 . √sin (x + a) + C (33) – 1 . sin–1 1 . cos2 2x + C sin a √sin x 4 3 (34) 5.√x2 + 4x + 10 – 7 log | x + 2 + √x2 + 4x + 10 | + C (35) x3 / 3 + C (36) 2 . [ (x + a) 3 / 2 – (x + b) 3 / 2 ] + C (37) –2 √1 – x + cos–1 √x + √x – x2 + C 3(a – b)

(38) x3 + x + C (39) 2 . tan5 √x + C (40) x – 1 . log | cos x – sin x | + C 3 5 2 2

(41) x – 1 . log | cos x + sin x | + C (42) 1 . log | 5x – 1 + √5 .√5x2 – 2x | + C 2 2 √5 (43) –1 . (1 + x2 ) 3 / 2. log 1 + 1 – 2 + C (44) tan–1( e

x ) + C

3x3 x2 3 (45) 3 log | 2 – sin x | + 4 + C (46) x – 5 log | x – 2 | + 10 log | x – 3 | + C 2 – sin x

(47) –1 tan–1 x + 2 tan–1 x + C (48) x + log | x | – 3 log | 1 – 2x | + C 3 3 2 2 4 (49) x + 2 . tan–1 x – 3 tan–1 x + C (50) x2 + 1 log | x + 1 | + 3 log | x – 1| + C √3 √3 2 2 2 2 (51) 1 log | x – 1 | – 1 log (x2 + 1) + 1 tan–1 x + C (52) – log |x – 1| + 1 log (x2 + 1) + tan–1 x + C 2 4 2 2 (53) 1 log x + 1 – 4 + C (54) 1 log x2 + 1 + C (55) 1 2 tan–1 x – tan–1 x +C 2 x – 1 x – 1 2 x2 + 3 6 2


(56) log ex – 1 + C (57) ( x – 1) .e x + C (58) e

x + C

ex x + 1 ( x – 1 )2

(59) ex + C (60) x log ( log x ) – x + C (61) ( 2x2 – 1).sin–1 x + x .√1 – x2 + C

1 + x log x 4 4

(62) (x2 + 1). tan–1 x – x + C (63) x.(sin–1 x)2 + 2 √1 – x2 . sin–1 x – 2x + C 2

(64) x tan x + log | cos x | + C (65) x2 2(log x )2 – 2 log x + 1 + C 4

(66) e 2 x . ( 2sin x – cos x ) + C (67) 2x tan–1 x – log (1 + x2 ) + C (68) x cos–1 x – √1 – x2 + C 5 2

(69) 2 . [ (2x – 1) . sin–1 √x + √x – x2 ] – x + C (70) ex tan x + C

π

(71) 2x + 3 .√x2 + 3x – 9 log | 2x + 3 + 2√x2 + 3x | + C 4 8 (72) 2x – 3 . √1 + 3x – x2 + 13 sin–1 2x – 3 + C 4 8 √13 (73) x – 4 .√x2 – 8x + 7 – 9 log | x – 4 + √x2 – 8x + 7 | + C 2 2

(74) x √x2 + 9 + 3 log | x + √x2 + 9 | + C 6 2

HOTS PROBLEMS

(1) tan x – cot x – 3x + C (2) (2 / 5) x + (3 / 5) x + C log (2 / 5) log (3 / 5) (3) log | cos x | + 1 log | cos 2x | – 1 log | cos 3x | + C (4) 1 . log | a cos2 x + b sin2 x | + C 2 3 2(b – a) (5) 1 . log sin (x – a) + C (6) 1 . log sin (x – b) + C cos (a – b) cos (x – b) sin (a – b) sin (a – x)

(7) 1 . log cos (x – a) + C (8) √푥 + 푥 + 1 . log (2x + 1) + 2√푥 + 푥 + C sin (a – b) cos (x – a) 2 (9) 2 | sin ( x / 2) – cos ( x / 2) | + C (10) – log cos x + 1 + √cos2x + cos x + C 2

(11) – √2 . log tan 3π – x + C (12) log 2sin x + 1 + 2 √sin2 x + sin x +C 4 4

(13) x cos 2a – sin2a log (x + a) + C (14) – cos a . sin–1(cosx. sec a) – sin a . log | sin x + √sin2 x – sin2 a | + C


(15) a.sin–1( x / a) + √a2 – x2 + C (16) 1 log sin 3x – 1 log sin 5x 3 5 (17) 2 (x + 1) 3 / 2 – 4 (x + 1) 1 / 2 + C (18) 2 [ (x + 1)3 / 2 – x 3 / 2 ] + C 3 3

(19) –1 . log 1 + 1 + C (20) x3 – x + 2tan–1 x + C n xn 3 (21) x – sin 4x + C (22) (πx + x2 ) + C 8 4 4

(23) log | sec2x + sec x . tan x | + C (24) x2 / 2 + C (25) x2 / 2 + C

(26) x2 / 2 + C (27) 4 tan 1 / 4 x + C (28) { log | ex + x

e | } / 2 + C

(29) log cos π – x + C (30) x 3a + 1 + a 3x + C (31) x + 4 log | x – 2 | + C 4 3a + 1 3 log a

(32) – ( x4 + 1 ) 1 / 4 / x + C (33) 1 log | x4 – 9 | + 1 log x2 – 3 + C (34) 1 log √3 + tan x + C 4 12 x2 + 3 2√3 √3 – tan x

(35) – 1 + C (36) log |1 + tan( x / 2) | + C 2(2tan x + 3)

(37) 1 log tan x + 1 tan2 x + 2 tan x + C (38) a2 sin–1 x2 + 1 √a4 – x4 + C 2 2 2 2 2 2 a2 2

(39) 1 log √3 + √5 tan ( x / 2 ) + C (40) 1 log √21 + tan (x / 2) + C √15 √3 – √5 tan ( x / 2 ) √21 √21 – tan (x / 2)

(41) tan–1 (tan (x / 2) + 1 + C (42) 1 log √65 + 4 sin x + 7cos x + C √65 4cos x + 7 sin x

(43) 1 log √74 + 5sin x + 7 cos x + C (44) 1 [ x + log | sin x – cos x | ] + C √74 6 cos x + 7 sin x 2 (45) 1 [ 23 x + 2 log | 4 sin x + 5 cos x | ] + C (46) 1 [ 28x – 3 log 2 sin x + 3 cos x ]+ C 41 13

(47) 1 6x + 3 log sin x + 2cos x + 3 – 8 tan–1 1 + tan (x / 2) + C (48) 1 log tan (x / 2) + 2 + C 5 2 5 tan (x / 2) – 3 (49) 1 tan–1 2 tan x + C (50) 1 tan–1(2 tan x) + C 2 3 2 (51) 1 . log 3 + sin x – cos x + tan –1 ( sin x + cos x ) + C 2√3 3 – sin x + cos x (52) 1 . log 3 + sin x – cos x – tan –1 ( sin x + cos x ) + C 2√3 3 – sin x + cos x (53) – 1 + C (54) cos–1( 2 cos x + 1) + C (55) log √ x2 + x + 1 – √x + C a tan x + b √ x2 + x + 1 + √x


(56) 1 log x5 + C (57) tan x + tan3 x + C (58) x cos a + sin a . log cos(x – a) + C 10 x5 + 2

(59) 1 log tan x – 2 + C (60) 7 tan–1 2x – 7 + √10 + 3x – x2 + C (61) 2√tan x + C 4 tan x + 2 2 7

(62) tan x + sec x + C (63) 2 tan–1 √ x2 + x + 1 + C (64) 1 log a2 sin2 x + b2 cos2 x + C √x a2 – b2

(65) log log log log x + C (66) –2√2 cos π + x – √2 log tan π + x + C 4 2 8 4

(67) 1 log tan x + 1 tan–1 a + C (68) 1 log x2 + √x4 + 4 + C √a2 – b2 2 2 b 2

(69) sin–1(sin x – cos x ) + C (70) –1 log 1 + 1 + C (71) –1 log 9x2 + 6x + 6 + C 2 1 + x2 30 (3x + 1)2

(72) x – log 3ex + 2 + 2 √e2x + 3 ex + 1 + C (73) 1 log 5 + 4sin x – 4 cos x + C 40 5 – 4sin x + 4 cos x

(74) 1 log √2 – 1 + tan (x / 2) – 1 + C 2√2 √2 + 1 – tan (x / 2) 2(sin x + cos x)

(75) 1 sin–1 √10 sin x + log | √10 . cos x + √10 cos2 x – 1 | + C √10 3 (76) sin x – sin3 x + 3 sin5 x – 1 sin7 x + C (77) 1 2x + 1 sin 2x – 1 sin 4x – cos 6x + C 5 7 32 2 2 (78) – cosec x – 2sin x + sin3 x + C (79) 1 [ (x2 + a2) 3 / 2 – (x2 – a2) 3 / 2 ] + C 3 6a2

(80) sin–1 2 x + C (81) – log log cot x + C

log 2 (82) 6 x5 / 6 + x2 / 3 + x1 / 2 + x1 / 3 + x1 / 6 + log | x1 / 6 – 1 | + C (83) 1 [ (x – 2)√1 – x2 – sin–1 x ] + C 5 4 3 2 2 (84) 1 log √2x + √x2 – 1 + C (85) – √x + a + C (86) √1 – x – cos–1 x + √1 – x2 + 2√2 √2x – √x2 – 1 a √x – a √1 + x

(87) secm x + C (88) log | sin x | – 2 sin2 x + 3 sin4 x – 2 sin6 x + 1 sin8 x + C m 4 3 8 (89) 1 log sin x + cos x – 1 + C (90) 1 log 1 + √2 – tan2 x + √2 log √2.cos x + √2cos2 x – 1 + C 2 sin x + cos x + 1 2 1 – √2 – tan2 x (91) – log log tan π + x + C (92) 1 sin x – 1 cos x + 1 log tan π + x + C 2 2 2 2 2√2 8 2

(95) 1 [ax + b log (a cos x + b sin x) ] + C (96) x3 sin–1 x + (x2 + 2)√1 – x2 + C a2 + b2 3 9

(97) x3 tan–1 x – x2 + 1 log(1 + x2) + C (98) 2 sin √x – 2√x .cos√x + C 3 6 6


(99) (2x – 1)sin–1√x + √x – x2 + C (100) (x + 1)tan–1√x – √x + C (101) x sec–1√x – √푥 − 1 + C 2 2

(102) x tan–1 √x – √ax + a tan–1 √x + C (103) sec x . tan x + 1 log sec x + tan x + C √a √a 2 2

(104) x [ sin( log x) – cos(log x) ] + C (105) x3 – x2 sin 2x – x cos 2x + sin 2x + C 2 6 4 4 8

(106) x tan 2x + 1 log cos 2x + C (107) x2 [ 2sin2 x – 1 ] + x sin 2x + cos 2x + C 2 4 4 4 8

(108) – x [ 2 cos 6x – 6 cos 2x – 3 cos 4x ] + sin 2x + sin 4x – sin 6x + C 48 4 16 36

(109) x3 log (1 + x) – x3 + x2 – x + 1 log(1 + x) + C (110) x ( log x – 1)2 + x + C 3 9 6 3 3

(111) ex . [ sin 4x – 4 cos 4x ] + C (112) x . e

x [ x + log x – 1 ] + C

68

(113) x n + 1 (log x)2 – 2 log x + 2 + C (114) 1 sin 2x . log tan π + x – log sec 2x + C (n + 1) (n + 1)2 (n + 1)3 2 4 (115), (117) 2x tan–1 x – log( 1 + x2 ) + C (116) 3x tan–1 x – 3 log( 1 + x2 ) + C 2

(118) – x cos4 x + 3x + sin 4x + sin 2x + C (119) x tan ( x / 2) + C 4 32 128 16

(120) e ax

[ a sin( bx + c ) – b cos( bx + c ) ] + C (121) x sin–1 x + log(1 – x2 ) – (sin–1 x )2 + C a2 + b2 √1 – x2 2 2

(122) sin x – x cos x + C (123) 2√1 + x [ log(1 + x) – 2 ] + C (124) ex [ cos 2x + 2sin 2x + 5] + C

x sin x + cos x 10

(125) 1 [ (2x2 + 3) sin 2x + 2x cos 2x ] + C (126) 1 [ xn sin( xn ) + cos( xn ) ] + C 4 n

(127) 2√x sin√x + cos√x + C (128) e m [ m cos – sin ] + C ; tan = x

1 + m2

(129) x log | x + √a2 + x2 | – √a2 + x2 + C (130) ex (x – 2) + C (131) ex + C

(x + 2) x + 2

(132) ex . log sec x + C (133) – e

x . cot x + C (134) ex. log( sec x + tan x) + C

(135) x + C (136) x sin( log x ) + C (137) x tan( log x ) + C 1 + log x

(138) ex + C (139) ex. log( 1 + cos 2x ) + C (140) ex. cot 2x + C

x2 2

(141) ex. log ( sin x ) + C (142) sin x √9 – sin2 x + 9 sin–1 sin x + C (143) x . ex + C

2 2 3 x + 2


(144) ex . log | x + √x2 + a2 | + C (145) x + 2 √x2 + 4x + 1 – √3 log | x + 2 + √x2 + 4x + 1 | + C

2 2

(146) 2x – 7 √7x – x2 – 10 + 9 sin–1 2x – 7 + C (147) 2x – 3 √(x – 1)(2 – x) + 1 sin–1 (2x – 3)+ C 4 8 3 4 8

(148) – 1 (1 + x – x2 ) 3 / 2 + 2x – 1 √1 + x – x2 + 5 sin–1 2x – 1 + C 3 4 16 √5

(149) 1 (x2 + x) 3 / 2 – 11 2x + 1 √x2 + x – 1 log x + 1 + √x2 + x + C 3 2 4 8 2

(150) (2x – 3) √x2 – 3x + 2 – 1 log 2x – 3 + 2 √x2 – 3x + 2 + C 4 8

(151) – 1 (3 – 4x – x2 ) 3 / 2 + (x + 2) √3 – 4x – x2 + 7 sin–1 x + 2 + C 3 2 2 √7

(152) 5 log | x – 1 | + 11 log | x – 3 | – 8 log | x – 2 | + C 2 2

(153) 1 tan–1 x – log | x + 1| + 1 log(x2 + 2) + C (154) x2 – x – 2 log | x – 2 | + log | x – 3 | + C √2 √2 2 2 (155) x2 + 3x – log | x – 1 | + 8 log | x – 2 | + C (156) x + 8 log | x – 1 | + 1 log | x + 2 | + C 2 3 3

(157) – x – log | x – 1 | + C (158) log | x | + 1 log 1 – x + C 2 1 + x

(159) 1 – log | x | + 1 log | x – 1 | + 3 log | x + 1 | – 1 + C 4 4 2(x – 1)

(160) log x + C (161) sec x + log √cos x – 1 + C (162) 1 sec2 (x / 2) + C √1 + x2 √cos x + 1 2 (163) 1 log tan x + C (164) log | sec x + tan x | – 2 tan( x / 2) + C 4 √4 + tan2 x

(165) – 1 log | 1 – cos x | – 1 log | 1 + cos x | + 2 log | 1 – 2cos x | + C 2 6 3

(166) 1 log 3 + 4sin x + C (167) log ex ( ex + 1 ) + C (168) 1 log 1 – sin x + 1 + C 5 2 + sin x (2ex + 1)2 4 1 + sin x 2(1 – sinx)

(169) 1 log | 1 + sin x | – 1 log | 1 – sin x | + 4 log | 5 – 4 sin x | + C 18 2 9

(170) 1 log | 1 – cos x | – 1 log | 1 + cos x | + 2 log | 3 + 2 cos x | + C (171) – x + 2 tan–1(2 tan x) + C 10 2 5 3 3

(172) 1 log √2 + sin x – cos x + 2 tan–1(sin x – cos x) + C (173) 1 – 1 + log x + C 3√2 √2 – sin x + cos x x 2x2 x + 1

(174) x2 + x + 1 log | x – 1 | – 1 log( x2 + 1) – 1 tan–1 x + C (175) 1 + 1 + 2 + 2 log x – 1 + C 2 2 4 2 3x3 x2 x x


(176) 1 + 1 + log x + C (177) 3 log x – 2 – 3 + 2 + C 2(x + 1)2 x + 1 x + 1 x – 1 (x – 2)2 x + 2 (178) 2 log x + 2 – 1 + C (179) x + log (sin x + cos x)3 + C (180) 1 tan–1 x2 – 1 + C x + 1 x + 1 (2 + sin 2x)2 √2 √2 . x

(181) 1 log x2 – √2x + 1 + C (182) 1 tan–1 x2 – 1 + 1 log x2 – √2x + 1 + C 2√2 x2 + √2x + 1 2√2 √2 . x 4√2 x2 + √2x + 1

(183) 1 tan–1 x2 – 1 – 1 log x2 – √2x + 1 + C (184) 1 tan–1 x2 – 1 + C 2√2 √2 . x 4√2 x2 + √2x + 1 √3 √3 . x

(185) 1 log x2 – x + 1 + C (186) 1 tan–1 x2 – 1 + 1 log x2 – x + 1 + C 2 x2 + x + 1 2√3 √3 . x 4 x2 + x + 1 (187) 1 tan–1 x2 – 1 – 1 log x2 – x + 1 + C (188) 1 tan–1 x2 – 9 + C 2√3 √3 . x 4 x2 + x + 1 3√2 3√2. x (189) 1 tan–1 x2 – 4 + C (190) 1 tan–1 tan2 x – 1 + C 2√2 2√2. x √2 √2 . tan x

(191) – 1 tan–1 cot x – 1 – 1 log cot x + √2 cot x + 1 + C √2 √2 cot x 2√2 cot x – √2 cot x + 1

(192) tan–1(√4x + 5 ) + C (193) log √2 + x – 1 + C (194) – 1 log √17 + √15 – 3x + C √2 + x + 1 √51 √17 – √15 – 3x

(195) 1 tan–1 x – 1 + 1 log x – √2x + 1 + C (196) 1 log √x + 4 – 2 – √x + 4 + C √2 √2x 2√2 x + √2x + 1 16 √x + 4 + 2 4x

(197) – 1 log x + 4 + √5x2 + 20 + C (198) – 1 log 2 √x2 + 1 – √3.x + C √5 x + 1 4√3 2√x2 + 1 + √3.x

(199) 1 log √x + 1 + 1 – √x + 1 + C (200) 1 log √1 + x2 – x + 1 log √1 + x2 – √2x + C 2 √x + 1 – 1 x 2 √1 + x2 + x √2 √1 + x2 + √2x

(201) 2√x + 2 + 2 log √x + 2 – √3 + C (202) 2 tan–1 x + C √3 √x + 2 + √3 √3 √3x + 3

(203) log x + √x2 – 1 – sec –1 x + C (204) –1 log √3 (x + 1) + 2√x2 + x + 1 + C √3 2√3(x – 1)

(205) 1 [ sin–1 ( x2 ) + √1 – x4 ] + C (206) tan–1 (tan2 x) + C 2

(207) 2 sin ( x / 2) – cos (x / 2) + √2 log sec π – x + tan π – x + C 4 2 4 2

(208) 1 2 tan–1 x2 – 4 – log x2 – √13 . x + 4 + C (209) 1 tan–1 x2 – 1 + C 16√3 √3. x x2 + √13 . x + 4 3 3x (210) 2 log sin2 x – 4 sin x + 5 + 7 tan–1 (sin x – 2) + C (211) 1 tan–1 tan x – 1 + 1 log tan x + √2 tan x + 1 + C √2 √2 tan x 2√2 tan x – √2 tan x + 1

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DEFINITE INTEGRALS 1. Evaluation :

If ∫f(x) dx = F(x) + C , b

Then a ∫f(x) dx = F(x) + C a b

a ∫f(x) dx = { F(b) + C } – { F(a) + C } b

a ∫f(x) dx = F(b) – F(a) ; with ‘a’ lower limit, ‘b’ upper limit.

2. Properties : a b b

(i) a ∫ f(x) dx = 0 (ii) a ∫ f(x) dx = a ∫ f(t) dt b a b b

(iii) a ∫ f(x) dx = – b ∫ f(x) dx (iv) a ∫ f(x) dx = a ∫ f(a + b – x) dx a a

(v) o ∫ f(x) dx = o ∫ f(a – x) dx b c b

(vi) a ∫ f(x) dx = a ∫ f(x) dx + c ∫ f(x) dx a a (vii) – a ∫ f(x) dx = 2 0∫ f(x) dx ; if f (–x) = f(x) , i.e, function is even . = 0 ; if f (–x) = – f(x) , i.e, function is odd. 2a a (viii) 0 ∫ f(x) dx = 2 0∫ f(x) dx ; if f (2a – x) = f(x) . = 0 ; if f (2a – x) = – f(x).

3. Definite Integral as Limit of a Sum : b

a ∫ f(x) dx = (b – a) Lim 1 . { f(a) + f(a + h) + f(a + 2h) + - - - + f(a + (n – 1)h )} n n

= (b – a) Lim 1 . {f(a + h) + f(a + 2h) + f(a + 3h) - - - + f(a + nh )} n n = Lim h .{f(a) + f(a + h) + f(a + 2h) - - - + f(a + (n – 1)h )} h 0 with , b – a = nh . 4. Special Summation :

Sin( x ) + Sin ( x + h ) + Sin( x + 2h ) + - - - + n terms = Sin( nh / 2 ) . Sin{x + ( n – 1)h / 2} Sin( h / 2) Cos( x ) + Cos ( x + h ) + Cos( x + 2h ) + - - - + n terms = Sin( nh / 2 ) . Cos{x + ( n – 1)h / 2} Sin( h / 2)


5. Some Important Summation : (i) ∑푘 = k + k + k + - - - + k ( up to n terms ) = nk

(ii) ∑푛 = 1 + 2 + 3 + - - - + n = ( )

(iii) ∑푛 = 1 + 2 + 3 + - - - + 푛 = ( )( )

(iv) ∑푛 = 1 + 2 + 3 + - - - + 푛 = ( )

(v) ∑푎. 푟 = 푎 + 푎푟 + 푎푟 + - - - + 푎. 푟 = ( )( – )

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PROBLEMS ON DEFINITE INTEGRAL 1. Problems on Limit of a Sum Prove the following integrals by limits of a sum, 5 3

*(1) 0∫ (x + 1) dx = 35 / 2 (2) 2∫ x2 . dx = 19 / 3

4 4

*(3) 1∫ (x2 – x ) dx = 27 / 2 *(4) 0∫ ( x + e2 x ) dx = ( e8 + 15) / 2 1 4

(5) – 1 ∫ ex dx = e – e – 1 *(6) 1∫ ( x 2 + x )dx = 85 / 2

3 b

(7) 2 ∫ x3 dx = 65 / 4 (8) a ∫ sin x dx = cos a – cos b b b

(9) a ∫ sin2 x dx = (b – a) – cos(b + a).sin(b – a) (10) a ∫ cos x dx = sin b – sin a π / 2 4

(11) 0 ∫ sin x dx = 1 (12) 1 ∫ {ex – sin x}dx = e4 – e + cos 4 – cos 1 b 3

(13) a ∫ {ex – ax }dx = eb – ea – (ab – aa )log a *(14) 1 ∫ {2x2 – 3x + 5}dx = 46 / 3 4 3

(15) 1 ∫ {ex – x2 } dx = e4 – e – 21 (16) 1 ∫ ax dx = ( a3 – a) log a


2. Problems on Properties of Definite Integrals :

1 π

(17) 0 ∫ x e x 2 dx (18) 0 ∫ {sin2( x / 2 ) – cos2( x / 2 )}dx

2 1

*(19) 1∫ 5x2 dx / ( x2 + 4x + 3 ) (20) – 1 ∫ 5x 4 .√x 5 + 1 dx 2 2

*(21) 0∫ dx / ( x + 4 – x2 ) (22) 1∫ {(1 / x) – ( 1 / 2x2 )} e 2 x . dx

1 2

*(23) 0∫sin–1 {2x / ( 1 + x2 )} dx *(24) 0∫ x.√x + 2 dx

/ 2 8

*(25) 0 ∫ ( 2 log sin x – log sin 2x ) dx . *(26) 2 ∫ | x – 5 | dx .

1

(27) 0 ∫ x sin x dx / (1 + cos 2 x) . (28) 0∫ dx / {√1 + x – √x }

π / 3 π / 2

*(29) π / 6 ∫ dx / ( 1 + √tanx ) *(30) 0 ∫ log (sin x) dx

5 π / 4

(31) – 5 ∫│x + 2│dx *(32) 0∫ log ( 1 + tanx ) dx

π 2 π

(33) 0∫ x dx / ( 1 + sinx ) (34) 0 ∫cos5x dx

a π / 2

(35) 0∫ √x dx / {√x + √a – x } (36) 0 ∫ cos2 x dx

3 / 2 π

*(37) – 1 ∫│x sin ( π x )│dx *(38) 0 ∫ x dx / ( a2 cos 2 x + b2 sin 2 x )

π / 2 π / 4

*(39) 0 ∫ ex ( 1 + sinx )dx / ( 1 + cosx ) *(40) 0∫ sin x cos x dx / (cos4 x + sin4 x )

π / 2 π / 2

*(41) 0 ∫ cos2 x dx / ( cos2 x + 4sin2 x ) *(42) π / 6∫ (sin x + cos x) dx / √sin 2x

π / 4 1

*(43) 0∫ (sin x + cos x) dx / {9 +16 sin (2x)} (44) 0∫ x ex dx

π 4

*(45) 0∫ x (tan x) dx / (sec x + tan x) *(46) 1 ∫ {│x – 1│+│x – 2│+│x – 3│}.dx

1 π

*(47) 0 ∫ tan–1{(2x – 1) / ( 1 + x – x2 )} dx *(48) 0 ∫ log ( 1 + cos x ) dx

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HOTS PROBLEMS Prove the followings : π 1

*(1) 0 ∫ x log (sin x) dx = – π2 log2 / 2 *(2) 0 ∫ log {sin (πx / 2)} dx = – log 2.

3 2

*(3) 0 ∫ [x] dx = 3 (4) 0∫ [ x 2] dx = 5 – √3 – √2 . π π / 2

(5) 0 ∫ sin3 x dx = 4 / 3. *(6) 0∫ sin4 x dx = 3π / 16

1 ∞

*(7) 0∫ log( 1 + x ) dx / ( 1 + x2 ) = ( π log2 ) / 8 *(8) 0 ∫ x dx / ( 1 +x )( 1 + x2 ) = π / 4 ∞ 1

(9) 0 ∫ log( 1 + x2 ) dx / ( 1 + x2 ) = π log 2 *(10) 0∫ cot – 1(1 – x + x2 ) dx = (π/2) – log2

1 π

(11) 0∫ log x dx / √1 – x2 = ( – π log2 ) / 2 (12) 0 ∫ 풙 풅풙(ퟏ − 풄풐풔 풂 . 풔풊풏 풙) =

흅(흅 풂)풔풊풏 풂

2 e

(13) 1∫ √x dx / {√x + √3 – x } = 1 / 2 (14) 1 / e ∫ │ log x │dx = 2 – ퟐ 풆

. π / 2 e

(15) 0∫ cos x dx / (1 + cos x + sin x) = (π / 4) – ( log2) / 2 (16) 1 ∫( log x )3 dx = 6 – 2e π / 2 π / 2

(17) 0 ∫ (cos x + sin x ) dx / ( 3 + sin 2x) = log √ퟑ (18) – π / 2 ∫ cos x dx / ( 1 + ex ) = 1

π / 2 1

*(19) 0 ∫ x dx / ( cos x + sin x) = π log ( √2 + 1) / 2√2 (20) – 1 ∫ log ( x + √1 + x2 ) dx = 0 π / 2 π

*(21) 0 ∫ dx / ( 3 + cos x + 2sin x) = tan–1 2 – ( π / 4 ). (22) 0 ∫dx / (a2 – 2a.cos x + 1) = π / (a2 –1)

π / 2 3

*(23) 0 ∫ sin2 x dx / (cos x + sin x) = {log (√2 + 1)} / √2 . *(24) –2 ∫ |1 – x2 |dx = 28 / 3

π 1

(25) 0 ∫ x. sin x dx / ( 1 + sin x) = π ( π – 2 ) / 2 . *(26) 0 ∫sin–1{2x / ( 1 + x2 )}dx = π / 2 – log2

6 1

*(27) 3 ∫ {│x – 3│+│x – 4│+│x – 5│}.dx = 19 / 2 . *(28) 0 ∫tan–1{2x / (1 – x2)}dx = π / 2 – log2

π / 2 π

(29) 0 ∫ sin 2x log ( tan x) dx = 0 (30) – π ∫│sin x│.dx = 4


π / 2 3π / 10

(31) 0 ∫ sin2 x dx / ( 1 + cos x . sin x) = π /3√3 *(32) π / 5 ∫sin x dx / (cos x + sin x) = π / 20.

π / 2 π / 2

(33) 0 ∫dx / ( 2cos x + 4sin x) = ퟏ√ퟓ

log ퟑ √ퟓퟐ

(34) – π / 2 ∫(cos│x│+ sin│x│)dx = 4

π π

(35) 0 ∫ x.tan x dx / sec x . cosec x = π2 / 4. *(36) 0 ∫x.tan x dx / (sec x + cos x) = π2 / 4

π 3 / 2

*(37) π / 2 ∫ ex (1 – sinx )dx / (1 – cosx ) = e π / 2 *(38) 0∫│x cos ( π x )│dx = (5π – 2) / 2π2

3 π / 2

*(39) Prove that, 0 ∫ | x3 – 3x2 + 2x |dx = 11 / 4 *(40) 0 ∫ |sin x – cos x | dx = 2√2 – 2. 2π 2π

*(41) Prove that, 0 ∫ sin6 x . cos5 x dx = 0 *(42) 0 ∫ dx / { e sin x + 1 } = π π / 2

(43) – π / 2 ∫dx / (cos│x│+ sin│x│) = 2√2log(√2 + 1) (44) Let f (x) = 2x + 1 ; 1 ≤ x ≤ 2 3

= x2 +1 ; 2 ≤ x ≤ 3 , show that , 1 ∫ f (x) dx = 34 / 3. (45) Let f (x) = 3x2 + 4 ; 0 ≤ x ≤ 2 4

= 9x – 2 ; 2 ≤ x ≤ 4 , show that , 0 ∫ f (x) dx = 66.

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ANSWERS

2. Problems on Properties of Definite Integrals : (17) e – 1 (18) 0 (19) 5 – 5 9 log 5 – log 3 (20) 4√2 2 2 4 2 3

(21) 1 . log 21 + 5 √17 (22) e2 (e2 – 2) (23) π – log 2 (24) 16 √2 (√2 + 1) √17 4 4 2 15

(25) π log 2 (26) 9 (27) π 2 (28) 4√2 (29) π (30) – π log 2 2 4 3 12 2

(31) 29 (32) π log 2 (33) π (34) 0 (35) a / 2 (36) π / 4 8

(37) 3 + 1 (38) π2 (39) e π / 2

(40) π (41) π (42) 2 sin–1 √3 – 1 π π2 2ab 8 6 2

(43) log 9 (44) 1 (45) π (π – 2) (46) 19 (47) 0 (48) – π log 2 40 2 2

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