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Find the integrals of the functions\[\frac{\cos x}{1+\cos x}\]

$\begin{array}{1 1}x-\tan \frac{x}{2}+c \\ x+\tan \frac{x}{2}+c \\ x-\cot \frac{x}{2}+c \\ x+\cot \frac{x}{2}+c \end{array} $

1 Answer

  • $1+\cos x=2cos^2\frac{x}{2}.$
  • $\cos x=2\cos^2\frac{x}{2}-1.$
  • $\int\sec^2x=\tan x+c.$
Given:$I=\int \frac{\cos x}{1+\cos x}$
Using the information from the tool box we get,
$\cos x=2\cos^2\frac{x}{2}-1$ and $1+\cos x=2\cos^2\frac{x}{2}.$
Now seperating the terms we get,
separating the terms we get,
$\;\;\;=\int dx-\frac{1}{2}\int \sec^2\frac{x}{2}dx.$
On integrating we get,
$\;\;\;=x-\tan x/2+c.$


answered Jan 30, 2013 by sreemathi.v