Question

The value of $\underset{0}{\overset{\pi }{\int }}{\cos }^{101}xdx$ is equal to

• A. $\frac{\pi }{4}$
• B. $\frac{1}{102}$
• C. ${\left(\frac{\pi }{3}\right)}^{101}$
• D. $0$

Answer 1

The correct option is D $0$

Let $I=\underset{0}{\overset{\pi }{\int }}{\cos }^{101}xdx\cdots \left(i\right)$

$\Rightarrow I=\underset{0}{\overset{\pi }{\int }}\left[\cos \right(\pi -x){]}^{101}dx$
$\Rightarrow I=\underset{0}{\overset{\pi }{\int }}-{\cos }^{101}xdx\cdots \left(ii\right)$
Adding $\left(i\right)$ and $\left(ii\right)$, we get $2I=0$
$\therefore I=0$