Integrals — Maths STD 12 Science — Question

Here f(x) = sin⁷x

$\therefore f\left( { - x} \right) = {\sin ^7}\left( { - x} \right)$

(-sin x)⁷

= -sin⁷x = -f(x)

$\therefore $ f(x) is an odd function of x.

$\therefore I = \int\limits_{\frac{{ - \pi }}{2}}^{\frac{\pi }{2}} {{{\sin }^7}xdx} = 0$

${\left[ {\because \int\limits_{ - a}^a {f\left( x \right)dx = 0} } \right.}$ when f(x) is an odd function]