Question

If $I={\int }_0^{\pi }{(\pi x-x^2)}^{100}\sin 2xdx$, then value of $I$ is?

• A. ${\pi }^{100}$
• B. $\frac{1}{2}({\pi }^{100}-{\pi }^{97})$
• C. $\frac{1}{2}({\pi }^{100}+{\pi }^{97})$
• D. $0$

Answer 1

The correct option is D $0$

$I={\int }_0^{\pi }{(\pi x-x^2)}^{100}\sin 2xdx={\int }_0^{\pi }{\left(\pi (\pi -x)\right)}^{100}\sin (2\pi -2x)dx=-{\int }_0^{\pi }{(\pi x-x^2)}^{100}\sin 2xdx\therefore 2I=0\Rightarrow I=0$