212 STEP 4. Review the Knowledge You Need to Score High
Example 9
Evaluate
∫
sinx− 1
cos
dx.
Rewrite
∫(
sinx
cosx
−
1
cosx
)
dx=
∫
(tanx −secx)dx=
∫
tanxdx−
∫
secxdx
=ln
∣∣
secx
∣∣
−ln
∣∣
secx+tanx
∣∣
+C=ln
∣
∣∣
∣
secx
secx+tanx
∣
∣∣
∣+C
or−ln
∣∣
sinx+ 1
∣∣
+C.
Example 10
Evaluate
∫
e^2 x
ex
dx.
Rewrite the integral as
∫
exdx=ex+C.
Example 11
Evaluate
∫
3
1 +x^2
dx.
Rewrite as 3
∫
1
1 +x^2
dx=3 tan−^1 x+C.
Example 12
Evaluate
∫
1
√
9 −x^2
dx.
Rewrite as
∫
1
√
32 −x^2
dx=sin−^1
(
x
3
)
+C.
Example 13
Evaluate
∫
7 xdx.
∫
7 xdx=
7 x
ln 7
+C
KEY IDEA
Reminder: You can always check the result by taking the derivative of the answer.
TIP • Be familiar with the instructions for the different parts of the exam before the day of
exam. Review the instructions in the practice tests provided at the end of this book.