Mathematics - Definite Integration Question with Solution | TestHub

$={\left[-x^n\cos x\right]}_0^{\pi }+n{\int }_0^{\pi }{x}^{n-1}\cos xdx$

$={\pi }^n+n{\left[x^{n-1}\sin x\right]}_0^{\pi }-n(n-1){\int }_0^{\pi }{x}^{n-2}\sin xdx$

$\Rightarrow I_n={\pi }^n+n\times 0-n(n-1)I_{n-2}$

Putting $n=5$, we get

$I_5={\pi }^5-20I_3$

$\Rightarrow I_5+20I_3={\pi }^5$