How to find the integral of sin^3[x]dx?

Immadvapsupvkd

Immadvapsupvkd

Answered question

2023-03-14

How to find the integral of sin 3 [ x ] d x ?

Answer & Explanation

unieventos8l9

unieventos8l9

Beginner2023-03-15Added 5 answers

sin 3 ( x ) d x = sin ( x ) ( 1 - cos 2 ( x ) ) d x
= sin ( x ) d x - sin ( x ) cos 2 ( x ) d x

For the first integral:
sin ( x ) d x = - cos ( x ) + C

Using substitution, compute the second integral as follows:
Let u = cos ( x ) d u = - sin ( x ) d x
Then
- sin ( x ) cos 2 ( x ) d x = u 2 d u
= u 3 3 + C
= 1 3 cos 3 ( x ) + C

Putting it all together, we get our final result:
sin 3 ( x ) d x = sin ( x ) d x - sin ( x ) cos 2 ( x ) d x
= - cos ( x ) + 1 3 cos 3 ( x ) + C

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