5 Steps to a 5 AP Calculus BC 2019

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

Rewrite the indefinite integral:

∫

csc^2 u

du

3

=−

1

3

cotu+c

=−

1

3

cot( 3 t)+c

∫π/ 2

π/ 4

csc^2 ( 3 t)dt=−

1

3

cot( 3 t)

]π/ 2

π/ 4

=−

1

3

[

cot

(

3 π

2

)

−cot

(

3 π

4

)]

=−

1

3

[ 0 −(− 1 )]=−

1

3

.

Verify your result with a calculator.

Example 3

Evaluate

∫e

1

lnt

t

dt.

Letu=lnt,du=

1

t

dt.

Rewrite:

∫

lnt

t

dt=

∫

udu=

u^2

2

+C=

(lnt)^2

2

+C

∫e

1

lnt

t

dt=

(lnt)^2

2

]e

1

=

(lne)^2

2

−

(ln 1)^2

2

=

1

2

− 0 =

1

2

.

Verify your result with a calculator.

Example 4

Evaluate

∫ 2

− 1

xe(x

(^2) + (^1) )
dx.
Letu=x^2 +1;du= 2 xdxor
dx
2
=xdx.
∫
xe(x
(^2) + (^1) )
dx=
∫
eu
du
2

=

1

2

eu+C=

1

2

e(x

(^2) + (^2) )
+C
Rewrite:
∫ 2
− 1
xe(x
(^2) + 1 )
dx=

1

2

e(x

(^2) + 1 )
] 2
− 1

=

1

2

e^5 −

1

2

e^2 =

1

2

e^2

(

e^3 − 1

)

.

Verify your result with a calculator.