Q112 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Mathematics

$$ \left(\int \frac{2 \cos x+1}{(2+\cos x)^2} d x\right)-\frac{\sin x}{2+\cos x}= $$

  • A
    $\frac{1}{2+\cos x}+C$
  • B
    $\sin x+C$
  • C
    $\frac{2}{2+\cos x}+C$
  • Correct answer
    D
    $\mathrm{C}$

Solution

Let $f(x)=\frac{\sin x}{2+\cos x}$ $$ \begin{aligned} f^{\prime}(x) & =\frac{(2+\cos x) \cos x-\sin x(-\sin x)}{(2+\cos x)^2} \\ & =\frac{2 \cos x+1}{(2+\cos x)^2} \end{aligned} $$ $\frac{\sin x}{2+\cos x}$ is the antiderivatives of $\frac{2 \cos x+1}{(2+\cos x)^2}$ $$ \begin{aligned} & \Rightarrow \int\left(\frac{2 \cos x+1}{(2+\cos x)^2}\right) d x=\frac{\sin x}{2+\cos x}+C \\ & \Rightarrow \int \frac{(2 \cos x+1)}{(2+\cos x)^2} d x-\frac{\sin x}{2+\cos x}=C \end{aligned} $$

Hence, the correct answer is option: 'D'.

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