integration of (sinx +x³)dx
Answers
Answer:
For the integration of a product of two functions (First function)*(Second function) =
f
(
x
)
,
∫
f
(
x
)
⋅
d
x
=
(
F
i
r
s
t
f
u
n
c
t
i
o
n
)
⋅
∫
(
sec
o
n
d
f
u
n
c
t
i
o
n
)
⋅
d
x
−
∫
(
d
d
x
(
F
i
r
s
t
f
u
n
c
t
i
o
n
)
⋅
∫
(
sec
o
n
d
f
u
n
c
t
i
o
n
)
⋅
d
x
.
This is called integration by parts.
Explanation:
The choice of first function and second function is arbitrary in case of most functions.
Here, we have
f
(
x
)
=
x
3
⋅
sin
x
and we will choose
x
3
as the first function and
sin
x
as the second.
Thus,
∫
f
(
x
)
⋅
d
x
=
∫
x
3
⋅
sin
x
⋅
d
x
⇒
∫
f
(
x
)
⋅
d
x
=
x
3
∫
sin
x
⋅
d
x
−
∫
(
d
x
3
d
x
⋅
∫
sin
x
⋅
d
x
)
⋅
d
x
=
−
x
3
⋅
cos
x
+
∫
3
x
2
⋅
cos
x
⋅
d
x
=
−
x
3
⋅
cos
x
+
3
[
x
2
∫
cos
x
⋅
d
x
−
∫
(
d
d
x
(
x
2
)
∫
cos
x
⋅
d
x
)
⋅
d
x
]
=
−
x
3
⋅
cos
x
+
3
[
x
2
sin
x
−
∫
2
x
sin
x
⋅
d
x
]
=
−
x
3
⋅
cos
x
+
3
[
x
2
sin
x
−
2
(
x
∫
sin
x
⋅
d
x
−
∫
(
d
d
x
(
x
)
∫
sin
x
⋅
d
x
)
⋅
d
x
]
=
−
x
3
cos
x
+
3
[
x
2
sin
x
−
2
(
−
x
cos
x
+
∫
cos
x
⋅
d
x
)
]
=
−
x
3
cos
x
+
3
[
x
2
sin
x
+
2
x
cos
x
−
2
sin
x
]
Since it is an indefinite integral, we add an arbitrary constant to it.
∫
x
3
sin
x
⋅
d
x
=
−
x
3
cos
x
+
3
x
2
sin
x
+
6
x
cos
x
−
6
sin
x
+
C
where
C
is the integration constant.
Explanation:
The choice of first function and second function is arbitrary in case of most functions.
Here, we have
f
(
x
)
=
x
3
⋅
sin
x
and we will choose
x
3
as the first function and
sin
x
as the second.
Thus,
∫
f
(
x
)
⋅
d
x
=
∫
x
3
⋅
sin
x
⋅
d
x
⇒
∫
f
(
x
)
⋅
d
x
=
x
3
∫
sin
x
⋅
d
x
−
∫
(
d
x
3
d
x
⋅
∫
sin
x
⋅
d
x
)
⋅
d
x
=
−
x
3
⋅
cos
x
+
∫
3
x
2
⋅
cos
x
⋅
d
x
=
−
x
3
⋅
cos
x
+
3
[
x
2
∫
cos
x
⋅
d
x
−
∫
(
d
d
x
(
x
2
)
∫
cos
x
⋅
d
x
)
⋅
d
x
]
=
−
x
3
⋅
cos
x
+
3
[
x
2
sin
x
−
∫
2
x
sin
x
⋅
d
x
]
=
−
x
3
⋅
cos
x
+
3
[
x
2
sin
x
−
2
(
x
∫
sin
x
⋅
d
x
−
∫
(
d
d
x
(
x
)
∫
sin
x
⋅
d
x
)
⋅
d
x
]
=
−
x
3
cos
x
+
3
[
x
2
sin
x
−
2
(
−
x
cos
x
+
∫
cos
x
⋅
d
x
)
]
=
−
x
3
cos
x
+
3
[
x
2
sin
x
+
2
x
cos
x
−
2
sin
x
]
Since it is an indefinite integral, we add an arbitrary constant to it.
∫
x
3
sin
x
⋅
d
x
=
−
x
3
cos
x
+
3
x
2
sin
x
+
6
x
cos
x
−
6
sin
x
+
C
where
C
is the integration constant.