Answers and Explanations 531
Answers
901–1,001
901.
1
2
ln csc
cot
ln sec
tan
x
x
x
x
C
−
−
+
)
(
+
Here’s the given integral:
cos
sin
sin( )
x
x
x
dx
−
∫
2
Begin by applying the identity sin(2x) = 2 sin x cos x to the denominator of the
integrand. Then split up the fraction and simplify:
cos
sin
sin( )
cos
sin
sin cos
cos
sin cos
s
x
x
x
dx
x
x
x
x
dx
x
x
x
−
=
−
=
−
∫
∫
2
2
1
2
i in
sin cos
sin
cos
csc
sec
x
x
x
dx
x
x
dx
x
x dx
)
(
=
−
(
)
=
−
(
)
∫
∫
∫
1
2
1
1
1
2
You can now apply basic antiderivative formulas to arrive at the solution:
1
2
1
2
(csc
sec )
ln csc
cot
ln sec
tan
x
x dx
x
x
x
x
C
−
=
−
−
+
)
(
+
∫
902.
1
4
4
(tan )
x
C
+
Here’s the given integral:
tan
cos
3
2
x
x
dx
∫
Begin by rewriting the integral:
tan
cos
tan sec
3
2
3
2
x
x
dx
x
x
∫
∫
=
Then use the substitution u = tan x so that du = sec
2
x
 
dx:
u du u
C
x
C
3
4
4
4
1
4
∫
=
+
=
+
(tan )
903.
1
4
1
4
2
1
8
2
2
x
x
x
x C
−
−
+
sin( )
cos( )
The given integral is
x
xdx
sin
2
∫
To integrate this function, use integration by parts: u dv uv vdu
=
− ∫
∫
. Let u = x so that
du = dx and let dv = sin
2 
x. Note that to find v by integrating dv, you have to use a
trigonometric identity: