Question

integration x sin³ x cos x dx​

Answer 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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