Evaluate: int frac{2 cos x+1}{(2+cos x)^2} d | vedclass

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(D) Let $f(x) = \frac{\sin x}{2+\cos x}$.Now,differentiating $f(x)$ with respect to $x$ using the quotient rule:$f'(x) = \frac{(2+\cos x)(\cos x) - \s

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$C$

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(D) Let $f(x) = \frac{\sin x}{2+\cos x}$.Now,differentiating $f(x)$ with respect to $x$ using the quotient rule:$f'(x) = \frac{(2+\cos x)(\cos x) - \sin x(-\sin x)}{(2+\cos x)^2}$$f'(x) = \frac{2 \cos x + \cos^2 x + \sin^2 x}{(2+\cos x)^2}$Since $\sin^2 x + \cos^2 x = 1$,we have:$f'(x) = \frac{2 \cos x + 1}{(2+\cos x)^2}$Thus,$\frac{\sin x}{2+\cos x}$ is the antiderivative of $\frac{2 \cos x+1}{(2+\cos x)^2}$.Therefore,$\int \frac{2 \cos x+1}{(2+\cos x)^2} d x = \frac{\sin x}{2+\cos x} + C$.Substituting this into the expression:$\int \frac{2 \cos x+1}{(2+\cos x)^2} d x - \frac{\sin x}{2+\cos x} = \left( \frac{\sin x}{2+\cos x} + C \right) - \frac{\sin x}{2+\cos x} = C$.

Evaluate: $\int \frac{2 \cos x+1}{(2+\cos x)^2} d x - \frac{\sin x}{2+\cos x}$

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