5 Steps to a 5 AP Calculus BC 2019

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

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.

8.

∫

xcosxdx

Answer:Letu=x,du=dx,dv=cosxdx, andv=sinx,

then

∫

xcosxdx=xsinx−

∫

sinxdx=xsinx+cosx+C.

9.

∫

5

(x+3)(x−7)

dx

Answer:

∫

5

(x+3)(x−7)

dx=

∫ (

− 1 / 2

x+ 3

+

1 / 2

x− 7

)

dx

=−

1

2

ln

∣∣

x+ 3

∣∣

+

1

2

ln

∣∣

x− 7

∣∣

+C=

1

2

ln

∣∣

∣∣x−^7

x+ 3

∣∣

∣∣+C

10.5 Practice Problems

Evaluate the following integrals in problems
1 to 25. 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