Part II: The Answers
530
Answers
801–900
Now use the substitution u = sin x, which gives you du = cos x
 
dx. You can also find the
new limits of integration. If x = π
2
, then u =
=
sin π
2
1, and if x = 0, then u = sin 0 = 0.
Using these values in the integral gives you
1
3
1 1
3
0 0
2
3
2
0
1
3
0
1
−
(
) = −






= − − −
(
)
=
∫ u du u u
(
)
899.
ln csc
cot
ln sin
x
x
x C
−
−
+
The given integral is
1 −
∫
cos
sin
x
x
dx
Start by rewriting the integrand as two terms:
1
1
sin
cos
sin
sin
cos
sin
csc
cos
sin
x
x
x
dx
x
dx
x
x
dx
x dx
x
x
d
−
(
)
=
−
=
−
∫
∫
∫
∫
x x
∫
Recall that csc
l n csc
cot
x dx
x
x C
=
−
+
∫
. To evaluate cos
sin
x
x
dx
∫
, use the substitution
u = sin x so that du = cos x
 
dx:
cos
sin
ln
ln sin
x
x
dx
u
du
u C
x C
∫
∫
=
=
+
=
+
1
Combining the antiderivatives of each indefinite integral gives you the solution:
csc
cos
sin
ln csc
cot
ln sin
x dx
x
x
dx
x
x
x C
−
=
−
−
+
∫
∫
900.
3
2
2
1
4
2
θ
θ
θ
−
−
+
cos
sin( ) C
Begin by expanding the integrand:
( sin )
s in
sin
1
12
2
2
+
=
+
+
(
)
∫
∫
θ
θ
θ
θ θ
d
d
Next, use the identity sin ( )
c os( )
2
1
2
1
2
θ
θ
=
−
(
) to produce the following integral:
1 2
1
2
1
2
3
2
2
1
2
2
3
2
2
+
+
−
(
) )
(
=
+
−
(
)
=
−
∫
∫
sin
c os( )
sin
cos( )
co
θ
θ
θ
θ
θ
θ
θ
d
d
s s
sin( )
θ
θ
−
+
1
4
2
C