Question

The value of $\underset{-\pi /4}{\overset{\pi /4}{\int }}$${\sin }^{103}x.{\cos }^{101}xdxis$

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

Answer 1

Answer: (D)

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