Calculus Chapter 6

6-1

Name ________________________________ Date ______________ Class ____________

Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties

In Problems 1–3, find each indefinite integral and check by differentiating.

1. 6x dx∫

Section 6-1 Antiderivatives and Indefinite Integrals

Theorem 1 Antiderivatives If the derivative’s of two functions are equal on an open interval (a, b), then the functions differ by at most a constant. Symbolically, if F and G are differentiable functions on the interval (a, b) and '( ) '( )F x G x= for all x in (a, b), then ( ) ( )F x G x k= + for some constant k. Formulas and Properties of Indefinite Integrals For C and k both a constant

1. 1

,1

nn x

x dx Cn

+= +∫ +

1n ≠ −

2. x xe dx e C= +∫

3. 1

ln ,dx x Cx

= +∫ 0x ≠

4. ( ) ( )kf x dx k f x dx=∫ ∫

5. [ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx± = ±∫ ∫ ∫

Calculus Chapter 6

6-2

2. 1

29x dx∫

3. 7 xe dx∫

4. Find all the antiderivatives for 17 4.dy

zdz

−= +

Calculus Chapter 6

6-3

In Problems 5–8, find each indefinite integral.

5. 3 2( 2 8)x x x dx+ −∫

6. 34

8x dx

x

⎛ ⎞−∫ ⎜ ⎟⎝ ⎠

7. 4

8 5xdx

x

+⎛ ⎞∫ ⎜ ⎟⎝ ⎠

Calculus Chapter 6

6-4

8. 5 6

6

5 3 xx x edx

x

⎛ ⎞+∫ ⎜ ⎟⎝ ⎠

In Problems 9–12, find the particular antiderivative of each derivative that satisfies the given conditions.

9. 3 2'( ) 4 6 3;R x x x= + + (1) 12R =

Calculus Chapter 6

6-5

10. 3 3 5;tdye t

dt= + − (0) 5y =

Calculus Chapter 6

6-6

11. 2

2

5 9;

dD x

dx x

−= (9) 50D =

Calculus Chapter 6

6-7

12. 1 2'( ) 6 7 ;h x x x− −= + (1) 3h =

Calculus Chapter 6

6-8

Calculus Chapter 6

6-9

Name ________________________________ Date ______________ Class ____________

Goal: To find the indefinite integrals using general indefinite integral formulas

Section 6-2 Integration by Substitution

Formulas: General Indefinite Integral Formulas

1. 1[ ( )]

[ ( )] '( ) ,1

nn f x

f x f x dx Cn

+= +∫ +

1n ≠ −

2. ( ) ( )'( )f x f xe f x dx e C= +∫

3. 1

'( ) ln ( )( )

f x dx f x Cf x

= +∫

4. 1

,1

nn u

u du Cn

+= +∫ +

1n ≠ −

5. u ue du e C= +∫

6. 1

lndu u Cu

= +∫

Definition: Differentials If ( )y f x= defines a differentiable function, then

1. The differential dx of the independent variable x is an arbitrary real number. 2. The differential dy of the dependent variable y is defined as the product of

'( )f x and dx: '( )dy f x dx= Procedure: Integration by Substitution

1. Select a substitution that appears to simplify the integrand. In particular, try to select u so the du is a factor in the integrand.

2. Express the integrand entirely in terms of u and du, completely eliminating the original variable and its differential.

3. Evaluate the new integral if possible. 4. Express the antiderivative found in step 3 in terms of the original variable.

Calculus Chapter 6

6-10

In Problems 1–8, find each indefinite integral and check the result by differentiating.

1. 2 3(6 3 5) (12 3)x x x dx+ − +∫

2. 3 23 ( 3 5)(3 3)x x x dx+ − +∫

Calculus Chapter 6

6-11

3. 3

4

2

2 16 1

tdt

t t

+∫

+ −

4. 0.09xe dx−∫

Calculus Chapter 6

6-12

5. 7( 7)x x dx+∫

6. 2 42(ln(3 ))x

dxx

∫

Calculus Chapter 6

6-13

7. 1

2

31

xe dxx

−∫

8. 2 3 512 (2 7)dy

x xdx

= +

Calculus Chapter 6

6-14

9. The indefinite integral can be found in more than one way. Given the integral, 2 22 ( 3) ,x x dx+∫ first use the substitution method to find the indefinite integral and

then find it without using substitution.

Calculus Chapter 6

6-15

Name ________________________________ Date ______________ Class ____________

Goal: To solve differential equations that involve growth and decay.

Section 6-3 Differential Equations; Growth and Decay

Theorem 1: Exponential Growth Law

If dQ

rQdt

= and 0(0) ,Q Q= then 0 ,rtQ Q e=

where 0Q = amount of Q at 0t =

r = relative growth rate (expressed as a decimal) t = time Q = quantity at time t Table 1: Exponential Growth

Description Model Solution Unlimited growth dy

dtky=

, 0k t > (0)y c=

kty ce=

Exponential decay dydt

ky= −

, 0k t > (0)y c=

kty ce−=

Limited growth ( )dydt

k M y= −

, 0k t > (0) 0y =

(1 )kty M e−= −

Logistic Growth ( )dydt

ky M y= −

, 0k t >

1(0) Mc

y +=

1 kMt

My

ce−=

+

Calculus Chapter 6

6-16

In Problems 1–4, find the general or particular solution, as indicated, for each differential equation.

1. 36dy

xdx

−=

2. 323 ;xdy

x edx

−= − (0) 4y =

Calculus Chapter 6

6-17

3. 6dy

ydx

= −

4. 4 ;dx

xdt

= (0) 2x =

5. Find the amount A in an account after t years if

0.07dA

Adt

= and (0) 8000A =

Calculus Chapter 6

6-18

6. A single injection o a drug is administered to a patient. The amount Q in the body decreases at a rate proportional to the amount present. For a particular drug, the rate

is 6% per hour. Thus, 0.06dQ

Qdt

= − and 0(0)Q Q= where t is time in hours.

a. If the initial injection is 5 milliliters [ (0) 5Q = ], find ( )Q Q t= satisfying both conditions.

b. How many milliliters (to two decimal places) are in the body after 8 hours? c. How many hours (to two decimal places) will it take for half the drug to be left in

the body?

Calculus Chapter 6

6-19

7. A company is trying to expose a new product to as many people as possible through radio ads. Suppose that the rate of exposure to new people is proportional to the number of those who have not heard of the product out of L possible listeners. No one is aware of the product at the start of the campaign, and after 15 days, 50% of L

are aware of the product. Mathematically, ( ),dN

k L Ndt

= − (0) 0,N = and

(15) 0.5 .N L= a. Solve the differential equation. b. What percent of L will have been exposed after 7 days of the campaign? c. How many days will it take to expose 75% of L?

Calculus Chapter 6

6-20

Calculus Chapter 6

6-21

Name ________________________________ Date ______________ Class ____________

Goal: To calculate the values of definite integrals using the properties.

Section 6-4 The Definite Integral

Theorem: Limits of Left and Right Sums If ( ) 0f x > and is either increasing or decreasing on [a, b], then its left and right sums approach the same real number as .n →∞ Theorem: Limit of Riemann Sums If f is a continuous function on [a, b], then the Riemann sums for f on [a, b] approach a real number limit I as .n →∞ Definition: Definite Integral Let f be a continuous function on [a, b]. The limit I of Riemann sums for f on [a, b] is

called the definite integral of f from a to b and is denoted as ( ) .ba f x dx∫

Properties: Definite Integrals

1. ( ) 0aa f x dx =∫

2. ( ) ( )b aa bf x dx f x dx= −∫ ∫

3. ( ) ( ) ,b ba akf x dx k f x dx=∫ ∫ k a constant

4. [ ( ) ( )] ( ) ( )b b ba a af x g x dx f x dx g x dx± = ±∫ ∫ ∫

5. ( ) ( ) ( )b c ba a cf x dx f x dx f x dx= +∫ ∫ ∫

Calculus Chapter 6

6-22

In Problems 1 and 2, calculate the indicated Riemann sum nS for the function 2( ) 17 2 .f x x= −

1. Partition [–1, 9] into five subintervals of equal length, and for each subinterval

1[ , ],k kx x− let 1( ) / 2.k k kc x x−= +

2. Partition [–4, 8] into four subintervals of equal length, and for each subinterval

1[ , ],k kx x− let 1(2 ) / 3.k k kc x x−= +

Calculus Chapter 6

6-23

In Problems 3–7, calculate the definite integral, given that

50 12.5x dx =∫ 5 2

0125

3x dx =∫ 7 2

5118

3x dx =∫

3. 50 3x dx∫

4. 5 20 (2 )x x dx+∫

Calculus Chapter 6

6-24

5. 7 20 2x dx∫

6. 05 7x dx∫

7. 7 27 ( 2 15)x x dx+ −∫

Calculus Chapter 6

6-25

Name ________________________________ Date ______________ Class ____________

Goal: To use the fundamental theorem of calculus to solve problems.

In Problems 1–7, evaluate the integrals.

1. 5 21 (6 2)x dx+∫

Section 6-5 The Fundamental Theorem of Calculus

Theorem 1: Fundamental Theorem of Calculus If f is a continuous function on [a, b], and F is any antiderivative of f, then

( ) ( ) ( )ba f x dx F b F a= −∫

Definition: Average Value of a Continuous Function f over [a, b]

1

( )ba f x dx

b a∫−

Calculus Chapter 6

6-26

2. 40 (7 )xe dx∫

3. 3 3 23 ( 2 8 7)x x x dx+ − +∫

4. 2213

5

3dx

x∫

+

Calculus Chapter 6

6-27

5. 2 2 3 61 2 (2 9)x x dx+∫

6. 2

21 3 2

3 2 5

5 3

x xdx

x x x

− +∫

− + −

Calculus Chapter 6

6-28

7. 2 21 3 6 1x x dx+∫

In Problems 8 and 9, find the average value of the function over the given interval.

8. 3( ) 4 8 2;f x x x= − + [2,8]

Calculus Chapter 6

6-29

9. 0.3( ) 2 xg x e−= [0,10]

10. The total cost (in dollars) of manufacturing x units of a product is ( ) 30,000 250 .C x x= +

a. Find the average cost per unit if 400 units are produced. [Hint: Recall that ( )C x is the average cost per unit.]

b. Find the average value of the cost over the interval [0, 400].

Calculus Chapter 6

6-30

11. A company manufactures a product and the research department produced the marginal cost function

'( ) 3004

xC x = − 0 800x≤ ≤

where '( )C x is in dollars and x is the number of units produced per month. Compute

the increase in cost going from a production level of 400 units per month to 800 units per month. Set up a definite integral and evaluate it.