Question

The value of ${\int }_{-\pi /4}^{\pi /4}{\sin }^{103}x.{\cos }^{101}xdx$ is

• A. ${\left(\frac{\pi }{4}\right)}^{103}$
• B. ${\left(\frac{\pi }{4}\right)}^{101}$
• C. $2$
• D. $0$

Answer 1

The correct option is D $0$

Using: If $f\left(x\right)$ is an odd function then ${\int }_{-a}^{a}f\left(x\right)=0$

Let $f\left(x\right)={\sin }^{103}x\cdot {\cos }^{101}x$

$\Rightarrow f(-x)=-{\sin }^{103}x{\cos }^{101}x=-f\left(x\right)$

$\Rightarrow f\left(x\right)$ is an odd function

Hence ${\int }_{-\pi /4}^{\pi /4}{\sin }^{103}x.{\cos }^{101}xdx=0$