JEE Main 2022 (Online) 28th July Evening ShiftQuestion 1
The value of the integral $\int_0^{\pi/2}60\frac{\sin(6x)}{\sin x}dx$ is equal to ________.
Definite Integration
Answer$104$
Solution
Concept. $\frac{\sin6x}{\sin x}$ is a sum of cosines (telescoping product-to-sum), which integrate directly. Formulas used. (1) $2\sin x\cos kx=\sin(k+1)x-\sin(k-1)x$Step 1 - Expand. By (1) with $k=1,3,5$, the sum telescopes: $$2\sin x\,(\cos x+\cos3x+\cos5x)=\sin6x.$$ So $\frac{\sin6x}{\sin x}=2(\cos x+\cos3x+\cos5x)$. Step 2 - Integrate. $$\int_0^{\pi/2}=2\Big(\sin\frac\pi2+\frac13\sin\frac{3\pi}2+\frac15\sin\frac{5\pi}2\Big)=2\Big(1-\frac13+\frac15\Big)=\frac{26}{15}.$$ Step 3 - Value. $60\cdot\frac{26}{15}=104$. Answer: 104