int frac{2+cos frac{x}{2}}{x+sin frac{x}{2}} | vedclass

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Question

(A) Let $I = \int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \,d x$. Substitute $t = x + \sin \frac{x}{2}$. Differentiating both sides with respect

Vedclass · Accepted Answer

$2 \log \left|x+\sin \frac{x}{2}\right|+c$,where $c$ is a constant of integration.

Vedclass · Answer

(A) Let $I = \int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \,d x$. Substitute $t = x + \sin \frac{x}{2}$. Differentiating both sides with respect to $x$,we get: $\frac{dt}{dx} = 1 + \cos \frac{x}{2} \cdot \frac{1}{2} = 1 + \frac{1}{2} \cos \frac{x}{2} = \frac{2 + \cos \frac{x}{2}}{2}$. Therefore,$(2 + \cos \frac{x}{2}) dx = 2 dt$. Substituting these into the integral: $I = \int \frac{2}{t} dt = 2 \int \frac{1}{t} dt$. Integrating,we get: $I = 2 \log |t| + c$. Substituting back $t = x + \sin \frac{x}{2}$: $I = 2 \log \left| x + \sin \frac{x}{2} \right| + c$.

$\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \,d x=$

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