Answers and Explanations 515
Answers
801–900
Use integration by parts with u = x so that du = dx and let dv = sin x
 
dx so that
v = –cos x:
x
xdx
x
x
x dx
x
x
x
sin
c os
cos
cos
sin
C
∫
∫
= −
+
= −
+
+
866.
−
+
−
+
x
x
x
x C
csc
ln csc
cot
Here’s the given integral:
x
x
x dx
csc cot
∫
Use integration by parts with u = x so that du = dx and let dv = csc x cot x
 
dx so that
v = –csc x:
x
x
x dx
x
x
xdx
x
x
x
x C
csc cot
csc
csc
csc
ln csc
cot
∫
∫
= −
+
= −
+
−
+
867.
x
x
x
x
C
ln(
)
ln
3 1
1
3
3 1
+ − +
+ +
The given integral is
ln(
)
3 1
x
dx
+
∫
Use integration by parts with u = ln(3x + 1) so that du
x
dx
=
+
3
3 1
and let dv = dx so that
v = x. Using the integration by parts formula gives you
ln(
)
ln(
)
3 1
3 1
3
3 1
x
dx
x
x
x
x
dx
+
=
+
(
) −
+
∫
∫
To evaluate the integral
3
3 1
x
x
dx
+
∫
, you can use long division to simplify, or you can
add 1 and subtract 1 from the numerator and split up the fraction:
ln(
)
ln(
)
ln(
)
3 1
3
3 1
3 1
3 1 1
3 1
3 1
3
x
x
x
x
dx x
x
x
x
dx
x
x
x
+
(
) −
+
=
+ −
+ −
+
=
+ −
∫
∫
+ +
+
−
+
(
)
=
+ −
−
+
(
)
=
+ − +
∫
∫
1
3 1
1
3 1
3 1
1
1
3 1
3 1
1
3
x
x
dx
x
x
x
dx
x
x
x
ln(
)
ln(
)
ln 3 3 1
x
C
+ +