Evaluate integration of Cos x/ 3 cos x + sin x dx.
Answers
Answered by
1
Answer:
Use a
u
-substitution to get
∫
sin
3
x
cos
x
d
x
=
sin
4
x
4
+
C
.
Explanation:
What we have in this integral is a function,
sin
x
, and its derivative,
cos
x
. That means the integral is solvable using a
u
-substitution:
Let
u
=
sin
x
→
d
u
d
x
=
cos
x
→
d
u
=
cos
x
d
x
With this substitution,
∫
sin
3
x
cos
x
d
x
becomes:
∫
u
3
d
u
This new integral is easily evaluated using the reverse power rule:
∫
u
3
d
u
=
u
3
+
1
3
+
1
+
C
=
u
4
4
+
C
Because
u
=
sin
x
, we can substitute to get a final answer of:
∫
sin
3
x
cos
x
d
x
=
sin
4
x
4
+
C
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