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Home  >>  CBSE XII  >>  Math  >>  Model Papers
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Evaluate : $ \int \large\frac{\sec^2x}{3+\tan\:x}$$dx $

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Toolbox:
  • Method of substitution :
  • Given f(x)dx can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
  • Put $x=g(t)$ so that $\large\frac{dx}{dt}$$=g'(t).$
  • $\Rightarrow dx=g'(t)dt.$
  • Thus $I=\int f(g(t).g'(t))dt.$
  • $\large\frac{d}{dx}$$(\tan x)=\sec^2xdx$
Step 1:
Let $I=\int \large\frac{\sec^2 x}{3+\tan x}$$dx$
Let $\tan x=t$
Differentiating with respect to $x$ we get,
$\sec^2 xdx=dt$
Step 2:
$\therefore I=\int\large\frac{dt}{3+t}$
$\quad\;\;=\log \mid 3+t\mid+c$
Substituting for $t$ we get,
$I=\log\mid 3+\tan x\mid+c$
answered Sep 21, 2013 by sreemathi.v
 
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