How do you find the antiderivative of $\int ({sin}^{7} x) d x$ $int (sin^7x)dx$ from $[- 1, 1]$ $[-1,1]$ ?

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Answer 1 · 2017-06-16T19:05:51

$\int_{- 1}^{1} {sin}^{7} x d x = 0 .$ $int_-1^1sin^7xdx=0.$

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

$\int_{a}^{a} f (x) d x = 0,$ $int_a^af(x)dx=0,$ if $f$ $f$ is an odd continuous function on $[- a, a] . (a > 0)$ $[-a,a]. (a>0)$

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

$f (x) = {sin}^{7} x \Rightarrow f (- x) = {sin}^{7} (- x) = - {sin}^{7} x = - f (x) .$ $f(x)=sin^7x rArr f(-x)=sin^7(-x)=-sin^7x=-f(x).$

$:. f$ is Odd.

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

How do you find the antiderivative of int (sin^7x)dx∫(sin7x)dxint (sin^7x)dx from [-1,1][−1,1][-1,1]?