Q80 / 180 · KCET 2016 · Mathematics
The value of \( \int_{-\Pi / 4}^{\Pi / 4} \sin ^{103} x \cdot \cos ^{101} x \mathrm{~d} \mathrm{x} \) is
- A\( \left(\frac{\Pi}{4}\right)^{103} \)
- B\( \left(\frac{\Pi}{4}\right)^{101} \)
- C\( 12 \)
- Correct answerD\( 00 \)
Solution
Given that $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{103} x \cdot \cos ^{101} x d x$
Let $f(x)=\sin ^{103} x \cos ^{101} x$
Then $f(-x)=\sin ^{103}(-x) \cos ^{101}(-x)$
$=-\sin ^{103} x \cos ^{101} x=-f(x)$
So, it is an odd function.
Since, $\int_{-a}^{a} f(x) d x=0$
Therefore $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{103} x \cdot \cos ^{101} x d x=0$
Hence, the correct answer is option: 'D'.