integral (e^x+3cosx-4x^3+2) dx
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Answer:
Let
t
=
x
2
. This implies that
d
t
=
(
2
x
)
d
x
. It may not seem like this is in the integrand, but note that
x
3
=
x
2
(
x
)
=
1
2
x
2
(
2
x
)
. Then:
I
=
1
2
∫
x
2
cos
(
x
2
)
(
2
x
)
d
x
I
=
1
2
∫
t
cos
(
t
)
d
t
Now we should do integration by parts, which comes in the form
∫
u
d
v
=
u
v
−
∫
v
d
u
. Let:
{
u
=
t
==⇒
d
u
=
d
t
d
v
=
cos
(
t
)
d
t
==⇒
v
=
sin
(
t
)
Then:
I
=
1
2
(
t
sin
(
t
)
−
∫
sin
(
t
)
d
t
)
I
=
t
sin
(
t
)
+
cos
(
t
)
2
I
=
x
2
sin
(
x
2
)
+
cos
(
x
2
)
2
+
C
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