int frac{sin frac{5x}{2}}{sin frac{x}{2}} dx | vedclass

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Question

(C) We need to evaluate $I = \int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx$.Multiply the numerator and denominator by $2 \cos \frac{x}{2}$:$I = \

Vedclass · Accepted Answer

$x + 2\sin x + \sin 2x + c$

Vedclass · Answer

(C) We need to evaluate $I = \int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx$.Multiply the numerator and denominator by $2 \cos \frac{x}{2}$:$I = \int \frac{2 \sin \frac{5x}{2} \cos \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} dx$Using the formula $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$:$I = \int \frac{\sin(3x) + \sin(2x)}{\sin x} dx$Using $\sin(3x) = 3 \sin x - 4 \sin^3 x$ and $\sin(2x) = 2 \sin x \cos x$:$I = \int \frac{3 \sin x - 4 \sin^3 x + 2 \sin x \cos x}{\sin x} dx$$I = \int (3 - 4 \sin^2 x + 2 \cos x) dx$Using $4 \sin^2 x = 2(1 - \cos 2x)$:$I = \int (3 - 2(1 - \cos 2x) + 2 \cos x) dx$$I = \int (1 + 2 \cos 2x + 2 \cos x) dx$Integrating term by term:$I = x + \sin 2x + 2 \sin x + c$.

$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx$ is equal to (where $c$ is a constant of integration).

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