Question

How do you find the antiderivative of `int (sin^7x)dx` from `[-1,1]`?

Answer 1

`int_-1^1sin^7xdx=0.`

Explanation:

We use the following Result to find the reqd. anti-derivative :

`int_a^af(x)dx=0,` if `f`is an odd continuous function on `[-a,a]. (a>0)`

The Proof of the Result is easy, so let us not bother about it.

`f(x)=sin^7x rArr f(-x)=sin^7(-x)=-sin^7x=-f(x).`

`:. f` is Odd.

Hence, by he Result, `int_(-1)^1sin^7xdx=0.`