Question

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{7}xdx$

Then $I=?$

Answer 1

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{7}xdx$ ______ (1)

Applying King's rule

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^7(\frac{\pi }{2}-\frac{\pi }{2}-x)dx$

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{7}xdx$ ________ (2)

$\left(1\right)+\left(2\right)\Rightarrow$

$2I=0$

$\Rightarrow I=0$