JEE Advanced 2014 · Paper 2, Mathematics question 42

Substitution. Let $e^u=\operatorname{cosec}x+\cot x$. Then $e^{-u}=\operatorname{cosec}x-\cot x$, so
$$\operatorname{cosec}x=\frac{e^u+e^{-u}}2,\qquad\frac{du}{dx}=-\operatorname{cosec}x.$$

Limits. $x=\frac\pi2$ gives $u=0$; $x=\frac\pi4$ gives $u=\log(1+\sqrt2)$.

Transform.
$$\int_{\pi/4}^{\pi/2}(2\operatorname{cosec}x)^{17}dx=\int_0^{\log(1+\sqrt2)}\frac{(e^u+e^{-u})^{17}}{\operatorname{cosec}x}du=\int_0^{\log(1+\sqrt2)}2(e^u+e^{-u})^{16}du.$$

(A)