Question

What is the antiderivative of `sinx^3`?

Answer 1

If you mean `(sinx)^3`, please see below. If you mean `sin(x^3)`, I can't help.

Explanation:

Here is one possibility

`int (sinx)^3 dx = int sin^2x sinx dx`

`= int ((1-cos^2 x) sinx dx`

`= int sinx dx + int underbrace(cos^2x)_u underbrace((-sinx) dx)_(du)`

`= -cosx + cos^3x/3 +C`

As is typical of expressions involving trigonometric functions, there are other ways to express this family of functions.

(WolframAlpha gives `1/12(cos(3x)-9cosx)+C` and simplifies the difference between our basic answers to `0`.)