Question

The value of ${\int }_0^{\pi }{\sin }^{120}x{\cos }^{101}xdx$ equals

• A. 0
• B. 1
• C. -1
• D. None of these

Answer 1

Answer: (A)

0

Let $f\left(x\right)={\int }_0^{\pi }{\sin }^{120}x{\cos }^{101}xdx$

$f(\pi -x)={\int }_0^{\pi }{\sin }^{120}(\pi -x){\cos }^{101}(\pi -x)dx$

$f(\pi -x)=-{\int }_0^{\pi }{\sin }^{120}x{\cos }^{101}xdx\dots \left(\sin \right(\pi -x)=\sin x)\left(\cos \right(\pi -x)=-cosx)$

$f(\pi -x)=-f\left(x\right)$

$\therefore f(\pi -x)=-f\left(x\right)$

$\therefore f\left(x\right)=0$ By using properties of definite integral
Ans: A