Created by T. Madas

Created by T. Madas

REDUCTION

FORMULAS

Created by T. Madas

Created by T. Madas

Question 1 (**)

21

0

12e

xnnI x dx

−= , n ∈� .

Show clearly that…

a) … ( ) 2

121 en nI n I

−

−= − − , 2n ≥ .

b) … 21

5

0

1 12 2e 8 13e

xx dx

− −= − .

proof

Created by T. Madas

Created by T. Madas

Question 2 (**)

0

2sinn

nI x dx

π

= , n ∈� .

Show clearly that…

a) … 2

1n n

nI I

n−

−= , 2n ≥ .

b) … ( )6 2

0

2 45sin 1 cos

256x x dx

π

π+ = .

proof

Created by T. Madas

Created by T. Madas

Question 3 (**)

( )

12

0

1 2 en x

nI x dx= − , n ∈� .

Show clearly that…

a) …. 12 1n nI nI −= − , 1n ≥ .

b) … 4 384 e 633I = − .

proof

Created by T. Madas

Created by T. Madas

Question 4 (**)

0

2cosn

nI x dx

π

= , n ∈� .

Show clearly that…

a) … 2

1n n

nI I

n−

−= , 2n ≥ .

b) … 5 2

0

2 8cos sin

105x x dx

π

= .

proof

Created by T. Madas

Created by T. Madas

Question 5 (**+)

( )e

1

lnn

nI x x dx= , n ∈� .

Show clearly that…

a) … 212 en nI nI −= − .

b) … ( )24

1e 3

4I = − .

proof

Created by T. Madas

Created by T. Madas

Question 6 (**+)

( )e

2

1

lnn

nI x x dx= , n ∈� .

a) Show clearly that

( )31

1e

3n nI nI −= − .

The part of the curve with equation ( )2

3 lny x x= , for 1 ex≤ ≤ , is rotated by 2π

radians about the x axis.

b) Show that the volume of the solid generated is given by

( )311e 89

π− .

proof

Created by T. Madas

Created by T. Madas

Question 7 (**+)

secnnI x dx= , n ∈� .

Show clearly that

( )( )2

2

1 2tan sec

1 1

nn n

nI x x I

n n

−

−

−= +

− −, 2n ≥ .

proof

Created by T. Madas

Created by T. Madas

Question 8 (***)

arsinh1

0

sinhnnI x dx= , n ∈� .

Show clearly that…

a) … ( ) 22 1n nnI n I −= − − , 2n ≥ .

b) … ( )5

17 2 8

15I = − .

proof

Created by T. Madas

Created by T. Madas

Question 9 (***)

The integral nI is defined for 0n ≥ as

4

0

tannnI x dx

π

= , n ∈� .

Show clearly that…

a) … 2

1

1n nI I

n−= −

−, 1n ≥ .

b) … ( )4

13 8

12I π= − .

proof

Created by T. Madas

Created by T. Madas

Question 10 (***)

0

tanh

an

nI x dx= , n ∈� .

Given that 1artanh2

a = , show clearly that

( )1

2

0.5

1

n

n nI In

−

−= −−

, 2n ≥ .

proof

Created by T. Madas

Created by T. Madas

Question 11 (***+)

By forming and using a suitable reduction formula, or otherwise, show that

21

5

0

2e 5e

2e

xx dx

− −= .

proof

Created by T. Madas

Created by T. Madas

Question 12 (***+)

The integral nI is defined for 0n ≥ as

0

2cosn

nI x x dx

π

= .

a) Show clearly that …

… ( ) 212

n

n nI n n Iπ

−

= − −

, 2n ≥ .

b) Hence find, in terms of π , exact expressions for …

i. … 4

4 2

0

2cos 3 24

16x x dx

ππ

π= − + .

ii. … 4

5 2

0

2 5sin 15 120

16x x dx

ππ

π= − +

44 2

0

2

cos 3 2416

x x dx

π

ππ= − + ,

45 2

0

2 5sin 15 120

16x x dx

π

ππ= − +

Created by T. Madas

Created by T. Madas

Question 13 (***+)

The integral nI is defined for 0n ≥ as

( )32

1

0

1nnI x x dx= − , n ∈� .

Show that

1

2

2 5n n

nI I

n−

=

+ , 1n ≥ ,

and use it to find as an exact fraction the value of 3I .

332

1155I =

Created by T. Madas

Created by T. Madas

Question 14 (***+)

ln 2

0

tanhnnI x dx= , n ∈� .

Show clearly that …

a) …

1

2

1 3

1 5

n

n nI In

−

−

= −

− , 2n ≥ .

b) …

2

1

1 3 5ln

2 5 4

r

r

r

∞

=

=

.

proof

Created by T. Madas

Created by T. Madas

Question 15 (****)

( )1

2

0

1n

nI x dx= − , n ∈� .

Show clearly that

1

2

2 1n n

nI I

n−=

+, 1n ≥ .

proof

Created by T. Madas

Created by T. Madas

Question 16 (****)

1

20

1

n

n

xI dx

x=

+ , n ∈� .

Show clearly that…

a) … ( ) 22 1n nnI n I −= − − , 2n ≥ .

b) … ( )1

3

20

12 2

31

xdx

x= −

+ .

proof

Created by T. Madas

Created by T. Madas

Question 17 (****)

( )3

2

0

3n

nI x dx= − , n ∈� .

Show clearly that…

a) … ( ) 12 1 6n nn I nI −+ = , 1n ≥ .

b) … ( )3

42

0

11523 3

35x dx− = .

proof

Created by T. Madas

Created by T. Madas

Question 18 (****)

12

0

1nnI x x dx= − , n ∈� .

Show clearly that…

a) … ( ) ( ) 22 1n nn I n I −+ = − , 2n ≥ .

b) …

12

0

161

315

nx x dx− = .

proof

Created by T. Madas

Created by T. Madas

Question 19 (****)

( )21n

nI x dx−

= + , n ∈� .

Show clearly that

( )2

1

1 2 1

2 2

n

n n

x x nI I

n n

−

+

+ −= + , 1n ≥ .

proof

Created by T. Madas

Created by T. Madas

Question 20 (****)

The integral nI is defined for 0n ≥ as

0

sinnnI d

π

θ θ θ= .

Show clearly that …

a) … ( ) 21nn nI n n Iπ −= − − , 2n ≥ .

b) … ( )4 4 2

0

2 1sin 2 12 48

32x x dx

π

π π= − + .

proof

Created by T. Madas

Created by T. Madas

Question 21 (****)

21

n

n

xI dx

x=

+ , n ∈� .

a) Find an expression for

( )121 2 1nd

x xdx

− +

.

b) Use part (a) to show that

( ) 1 221 1n

n nnI n I x x−

−+ − = + , 2n ≥ .

( ) ( ) ( ) ( )1 1 12 2 21 2 2 2 21 1 1 1n n nd

x x n x x x xdx

−− −

+ = − + + +

Created by T. Madas

Created by T. Madas

Question 22 (****+)

2e sinx nnI x dx≡ , n ∈� , 2n ≥ .

Use integration by parts twice to show

( ) ( ) ( )2 2 124 1 2sin cos e sinx n

n nn I n n I x n x x−

−+ = − + − .

SPX-G , proof

Created by T. Madas

Created by T. Madas

Question 23 (****+)

Find a suitable reduction formula and use it to find

( )1

10

0

lnx x dx .

You may assume that the integral converges.

Give the answer as the product of powers of prime factors.

( )1

10 3 4 2

0

ln 2 3 5 7x x dx−

= × × ×

Created by T. Madas

Created by T. Madas

Question 24 (****+)

sin

sinn

nxI dx

x= , n ∈� .

a) Show by considering 2n nI I+ − that

( )2

2sin 1

1n nI I n x C

n+ = + + +

+, 0n ≥ .

b) Show further that

( )3

4

sin 6 112 3 17 2

sin 15

xdx

x

π

π= − .

proof

Created by T. Madas

Created by T. Madas

Question 25 (****+)

( )

0

sin

sinn

nI d

πθ

θθ

= .

The integral above is defined for positive integer values n .

a) Use trigonometric identities to show that

( ) ( )( )

sin sin 22cos 1

sin

n nn

θ θθ

θ

− − = − .

b) Hence show that

2n nI I −= , 2n ≥ .

c) Evaluate nI in both cases, where n is either odd or even positive integer.

0 if is even

if is oddn

nI

nπ

=

Created by T. Madas

Created by T. Madas

Question 26 (****+)

3

0

3e tanx n

nI x dx

π

= , n ∈� .

a) Show clearly that…

i. … ( )1 1e 3 3n

n n nnI I nIπ

+ −= − − , 1n ≥ .

ii. … 0 4 3 13I I I I= + − .

b) Hence find the exact value of

( )3 3 2

0

3e tan tan sec 4x

x x x dx

π

+ − .

proof

Created by T. Madas

Created by T. Madas

Question 27 (****+)

2

,0

sin cosm nm nI d

π

θ θ θ= , m ∈� , n ∈� .

a) Show clearly that

, 2,

1m n m n

mI I

m n−

−=

+.

b) Hence find an exact value for

22

0

sin sin 2 d

π

θ θ θ .

128

315

Created by T. Madas

Created by T. Madas

Question 28 (*****)

It is given that

12

0

an

nI x a x dx+

= − , n ∈� , 0n ≥

where a is a positive constant.

a) Use integration by parts to show

02 2

4 1

n

n

na II

n n

+ =

+ , 1n ≥ .

b) Determine the value of

210 2

0

4x x dx− .

SPX-E , 42π

Created by T. Madas

Created by T. Madas

Question 29 (*****)

It is given that

( )1

,0

1n m

n mI x x dx= − ,

where ,n m∈� , with , 0n m ≥ .

a) Show that …

i. … , 1, 1, 1n m n m n mI I I− − +− = − .

ii. … , 1, 1n m n m

nI I

m− +=

b) Hence derive an expression of ,n mI and use it to find

( )12

13

0

7 1x x dx− .

SPX-C , ( )12

13

0

327 1

45x x dx− =

Created by T. Madas

Created by T. Madas

Question 30 (*****)

( ) ( ) ( ),

b

m n

a

I m n b x x a dx= − − , m∈� , n∈� .

Show that

( )( )

( )1! !

,1 !

m nm nI m n b a

m n

+ += −

+ +,

where a and b are real constants such that b a>

SPX-B , proof

Created by T. Madas

Created by T. Madas

Question 31 (*****)

2 20

an

n

xI dx

a x=

− , n ∈� , 0a > .

Show clearly that…

a) … ( )2

2

1n n

a nI I

n−

−= , 2n ≥ .

b) …

43 2

22

3 18 36 183 16

4

x x xdx

x xπ

− + −= −

− .

FP4-S , proof