5 Steps to a 5 AP Calculus BC 2019

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

11.5 Rapid Review

1. Evaluate

∫x

π/ 2

costdt.

Answer: sint]xx/ 2 =sinx−sin

(

π

2

)

=sinx−1.

2. Evaluate

∫ 1

0

1

x+ 1

dx.

Answer: ln(x+1)]^10 =ln 2−ln 1=ln 2.

3. IfG(x)=

∫x

0

(2t+1)^3 /^2 dt, findG′(4).

Answer: G′(x)=(2x+1)^3 /^2 andG′(4)= 93 /^2 =27.

4. If

∫k

1

2 xdx=8, findk.

Answer: x^2

]k

1 =^8 ⇒k

(^2) − 1 = 8 ⇒k=±3.

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)].

7.

∫∞

1

dx

x^2

Answer:

∫∞

1

dx

x^2

=limn→∞

∫n

1

dx

x^2

=limn→∞

[

− 1

x

]n

1

=limn→∞

[

− 1

n

+ 1

]

=1.

8.

∫ 1

0

dx

√

x

Answer:

∫ 1

0

√dx

x

=limk→ 0 +

∫ 1

k

√dx

x

=limk→ 0 +

[

2

√

x

] 1

k=klim→ 0 +

[

2 − 2

√

k

]

=2.