Answers and Explanations 525
Answers
801–900
1
2
1
2
1
2
1
2
2
1
2 4
1
2
2
0 1
2
0
4
0
4
+
(
)
=
+
(
)
=
+
( ) − +
∫
cos( )
sin( )
sin
θ
θ
θ
θ
π
π
π
π
d
s sin( )
0
8
1
4
(
)
= +
π
886.
tan x x C
− +
Here’s the given integral:
tan
2 x dx
∫
As long as you remember your fundamental trigonometric identities, this problem is an
easy one!
tan
s ec
tan
2
2
1
x dx
x
dx
x x C
∫
∫
=
−
(
)
=
− +
887.
tan
tan
t
t C
+
+
1
3
3
The given integral is
sec
4 t dt
∫
Begin by factoring out sec
2
t and using a trigonometric identity on one sec
2
t to write it
in terms of tan t:
sec
s ec sec
tan
sec
4
2
2
2
2
1
t dt
t
t dt
t
tdt
∫
∫
∫
=
=
+
(
)
Now use the substitution u = tan t so that du = sec
2
t
dt. Using these values in the
integral gives you
1
1
1
3
1
3
2
2
2
3
3
+
(
)
=
+
(
)
= +
+
=
+
+
∫
∫
tan
sec
tan
tan
t
tdt
u du
u
u C
t
t C
Answers
801–900
1
2
1
2
1
2
1
2
2
1
2 4
1
2
2
0 1
2
0
4
0
4
+
(
)
=
+
(
)
=
+
( ) − +
∫
cos( )
sin( )
sin
θ
θ
θ
θ
π
π
π
π
d
s sin( )
0
8
1
4
(
)
= +
π
886.
tan x x C
− +
Here’s the given integral:
tan
2 x dx
∫
As long as you remember your fundamental trigonometric identities, this problem is an
easy one!
tan
s ec
tan
2
2
1
x dx
x
dx
x x C
∫
∫
=
−
(
)
=
− +
887.
tan
tan
t
t C
+
+
1
3
3
The given integral is
sec
4 t dt
∫
Begin by factoring out sec
2
t and using a trigonometric identity on one sec
2
t to write it
in terms of tan t:
sec
s ec sec
tan
sec
4
2
2
2
2
1
t dt
t
t dt
t
tdt
∫
∫
∫
=
=
+
(
)
Now use the substitution u = tan t so that du = sec
2
t
dt. Using these values in the
integral gives you
1
1
1
3
1
3
2
2
2
3
3
+
(
)
=
+
(
)
= +
+
=
+
+
∫
∫
tan
sec
tan
tan
t
tdt
u du
u
u C
t
t C