5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:57

Def inite Integrals 263

5. IfG(x) is an antiderivative of (ex+1) andG(0)=0, findG(1).
   Answer: G(x)=ex+x+C
   G(0)=e^0 + 0 +C= 0 ⇒C=−1.
   G(1)=e^1 + 1 − 1 =e.


6. IfG′(x)=g(x), express

∫ 2

0

g(4x)dxin terms ofG(x).

Answer: Letu= 4 x;

du

4

=dx.

∫

g(u)

du

4

=

1

4

G(u).Thus,

∫ 2

0

g( 4 x)dx=

1

4

G( 4 x)

] 2

0

=

1

4

[G(8)−G(0)].

12.5 Practice Problems

Part A The use of a calculator is not

allowed.

Evaluate the following definite integrals.

1.

∫ 0

− 1

(1+x−x^3 )dx

2.

∫ 11

6

(x− 2 )^1 /^2 dx

3.

∫ 3

1

t

t+ 1

dt

4.

∫ 6

0

∣∣

x− 3

∣∣

dx

5. If

∫k

0

(6x−1)dx=4, findk.

6.

∫π

0

sinx

√

1 +cosx

dx

7. Iff′(x)=g(x) andgis a continuous
   function for all real values of∫ x, express
   2
   1

g(4x)dxin terms off.

8.

∫ln 3

ln 2

10 exdx

9.

∫e 2

e

1

t+ 3

dt

10. If f(x)=

∫x

−π/ 4

tan^2 (t)dt, findf′

(

π

6

)

.

11.

∫ 1

− 1

4 xex

2

dx

12.

∫π

−π

(

cosx−x^2

)

dx

Part B Calculators are allowed.

13. Findkif

∫ 2

0

(

x^3 +k

)

dx= 10.

14. Evaluate

∫ 3. 1

− 1. 2

2 θcosθdθto the nearest

100th.

15. Ify=

∫x 3

1

√

t^2 + 1 dt, find

dy

dx

.

16. Use a midpoint Riemann sum with four
    subdivisions of equal length to find the

approximate value of

∫ 8

0

(

x^3 + 1

)

dx.

17. Given

∫ 2

− 2

g(x)dx= 8

and

∫ 2

0

g(x)dx=3, find