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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Find the integrals of the functions\[\frac{\cos2x+2\sin^2x}{\cos^2x}\]

$\begin{array}{1 1}\tan x + c \\\cot x + c \\ \sin x + c \\ \cos x + c \end{array} $

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1 Answer

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Toolbox:
  • $(i)\cos2x=1-2\sin^2 x+c.$
  • $(ii)\;\int \sec^2xdx=\tan x+c$.
Given $I=\int\frac{\cos2x+2\sin^2x}{\cos^2x}dx.$
 
This can be written as ($2\sin^2x=1-\cos 2x$).
 
$I=\int\frac{\cos2x+(1-\cos 2x)}{\cos^2x}dx.$
 
On cancelling cos 2x we get,
 
$\;\;\;=\int\frac{1}{\cos^2x}dx=\int \sec^2xdx.$
 
On integrating we get,
 
$\int\frac{\cos 2x+2\sin^2x}{\cos ^2x}dx=\tan x+c.$

 

answered Feb 3, 2013 by sreemathi.v
 
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