Q52 / 150 · MHT CET 2007 · Mathematics

$\int \frac{x+\sin x}{1+\cos x} d x$ is equal to

  • Correct answer
    A
    $x \tan \frac{x}{2}+c$
  • B
    $\log (1+\cos x)+c$
  • C
    $\cot \frac{x}{2}+c$
  • D
    $\log (x+\sin x)+c$

Solution

Let $I=\int \frac{x+\sin x}{1+\cos x} d x$ $$ \begin{array}{l} =\int\left(\frac{x}{2} \sec ^{2} \frac{x}{2}+\tan \frac{x}{2}\right) d x \\ =x \tan \frac{x}{2}-\int \tan \frac{x}{2} d x+\int \tan \frac{x}{2} d x+c \end{array} $$ $=x \tan \frac{x}{2}+c$

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

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