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

MA 3972-MA-Book May 8, 2018 13:52

242 STEP 4. Review the Knowledge You Need to Score High

Answer:Rewrite as

∫

x(x^2 −1)^1 /^2 dx. Letu=x^2 −1.

Thus,

du

2

=xdx⇒

1

2

∫

u^1 /^2 du=

1 u^3 /^2

23 /^2

+C=

1

3

(x^2 −1)^3 /^2 +C.

4. Evaluate

∫

sinxdx.

Answer:−cosx+C.

5. Evaluate

∫

cos(2x)dx.

Answer:Letu= 2 xand obtain

1

2

sin 2x+C.

6. Evaluate

∫

lnx

x

dx.

Answer:Letu=lnx;du=

1

x

dxand obtain

(lnx)^2

2

+C.

7. Evaluate

∫

xex^2 dx.

Answer:Letu=x^2 ;

du

2

=xdxand obtain

ex^2

2

+C.

11.4 Practice Problems

Evaluate the following integrals in problems

1 to 20. No calculators are allowed. (However,

you may use calculators to check your

results.)

1.

∫

(x^5 + 3 x^2 −x+1)dx

2.

∫ (√

x−

1

x^2

)

dx

3.

∫

x^3 (x^4 −10)^5 dx

4.

∫

x^3

√

x^2 + 1 dx

5.

∫

x^2 + 5

√

x− 1

dx

6.

∫

tan

(

x

2

)

dx

7.

∫

xcsc^2 (x^2 )dx

8.

∫

sinx

cos^3 x

dx

9.

∫

1

x^2 + 2 x+ 10

dx

10.

∫

1

x^2

sec^2

(

1

x

)

dx

11.

∫

(e^2 x)(e^4 x)dx

12.

∫

1

xlnx

dx

13.

∫

ln(e^5 x+^1 )dx

14.

∫

e^4 x− 1

ex

dx

15.

∫

(9−x^2 )

√

xdx

16.

∫ √

x

(

1 +x^3 /^2

) 4

dx