integrate x cube cos square x dx
Answers
Answer:
1
2
x
2
sin
(
x
2
)
+
1
2
cos
(
x
2
)
+
C
Explanation:
We can't just integrate straight away, so we try substitution.
While trying substitution, we observe that we could integrate
cos
(
x
2
)
x
d
x
by substitution.
So, let's split the integrand and use integration by parts.
∫
x
3
cos
(
x
2
)
d
x
=
∫
x
2
cos
(
x
2
)
x
d
x
(We notice that we could rewrite this as
∫
u
cos
(
u
)
1
2
d
u
, but we don't see how to integrate that, so we'll continue with parts for now.)
∫
x
2
cos
(
x
2
)
x
d
x
Let
u
=
x
2
and
d
v
=
cos
(
x
2
)
x
d
x
.
Clearly
d
u
=
2
x
d
x
, and
we can integrate
d
v
by substitution to get.
1
2
sin
(
x
2
)
.
u
v
−
∫
v
d
u
=
1
2
x
2
sin
(
x
2
)
−
∫
x
sin
(
x
2
)
d
x
Integrate by substitution agan to finish.
∫
x
3
cos
(
x
2
)
d
x
=
1
2
x
2
sin
(
x
2
)
+
1
2
cos
(
x
2
)
+
C
Check the answer by differentiating.