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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate :$\int \large\frac{1}{1+\cos x}$$dx$

\[\begin {array} {1 1} (a)\;\tan \bigg(\frac{x}{2}\bigg)+c \\ (b)\;\sec^2 \bigg(\frac{x}{2}\bigg)+c \\ (c)\;\frac{1}{2}\tan \bigg(\frac{x}{2}\bigg)+c \\ (d)\;2.\tan \bigg(\frac{x}{2}\bigg)+c \end {array}\]

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A)
Using trigonometry property:-
$1+ \cos 2 \theta = 2\cos ^2 \theta$
=> $ \large\frac{1}{2 \cos ^2 x/2}$
=> $\large\frac{1}{2} $$\int \sec^2 \bigg(\large\frac{x}{2}\bigg) $$dx$
=> $\large\frac{1}{2} \times $$ \tan (\large \frac{x}{2})$$ \times 2 +c$
By solving this we get,
$\tan \bigg(\large\frac{x}{2}\bigg)+$$c$
Hence a is the correct answer.
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