Question

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

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

Answer 1

Answer: (D)

Let $I=\int\limits_{-\pi / 4}^{\pi / 4} \sin ^{103} x \cdot \cos ^{101} x\,dx$
Let $ f(x) =\sin ^{103} x \cdot \cos ^{101} x$
$ \Rightarrow f(-x) =\sin ^{103}(-x) \cos ^{101}(-x) $
$=-\sin ^{103} x \cos ^{101} x=-f(x) $
$ \therefore I =0 $
$\left[\because \int\limits_{-a}^{a} f(x) d x=0\right.$ if $f(x)$ is odd, i. e. $f(-x)=-f(x)]$