JEE Main 2023 (Online) 24th January Morning ShiftQuestion 1
The value of $\frac8\pi\int_0^{\frac\pi2}\frac{\cos^{2023}x}{\sin^{2023}x+\cos^{2023}x}dx$ is ______.
Definite Integration
Answer$2$
Solution
Concept. $x\to\frac\pi2-x$ swaps $\sin$ and $\cos$; the two forms add up to $1$. Formulas used. (1) $\int_0^ag(x)\,dx=\int_0^ag(a-x)\,dx$Step 1 - Use (1). $$I=\int_0^{\pi/2}\frac{\sin^{2023}x}{\cos^{2023}x+\sin^{2023}x}dx.$$ Step 2 - Add. $2I=\int_0^{\pi/2}1\,dx=\frac\pi2$, so $I=\frac\pi4$. Step 3 - Value. $\frac8\pi\cdot\frac\pi4=2$. Answer: 2