JEE Main 2021 (Online) 4th August Morning ShiftQuestion 1
If $I_1=\int_{0}^{\pi}xe^{\operatorname{cosec}x}dx$ and $I_2=\int_{0}^{\pi}e^{\operatorname{cosec}x}dx$, then the ratio $I_1:I_2$ is equal to
Definite Integration
Solution
Concept. $\operatorname{cosec}(\pi-x)=\operatorname{cosec}x$, so King's rule turns the factor $x$ into $\pi-x$. Formulas used. (1) $\int_0^\pi g(x)\,dx=\int_0^\pi g(\pi-x)\,dx$Step 1 - Use (1). $$I_1=\int_0^\pi(\pi-x)e^{\operatorname{cosec}x}dx=\pi I_2-I_1.$$ Step 2 - Ratio. $2I_1=\pi I_2$, so $I_1:I_2=\pi:2$. Answer: (A)