$\int\limits_{0}^{2\pi} \text{cosec}^7 x \, dx =$
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
We know, $\int\limits_{0}^{2a} f(x) dx = 0$, if $f(2a-x) = -f(x)$
Let $f(x) = \text{cosec}^7 x$.
Now, $f(2\pi - x) = \text{cosec}^7(2\pi - x)$
$= -\text{cosec}^7 x = -f(x)$
$∴\int\limits_{0}^{2\pi} \text{cosec}^7 x \, dx = 0$