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Integrate the function $\sec^2(7-4x)$

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Toolbox:
  • (i)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 $\frac{dx}{dt}=g'(t).$
  • $\Rightarrow $dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
  • (ii)$\int \sec^2x=\tan x+c.$
Given $I=\int \sec^2(7-4x).$
 
Put 7-4x=t.
 
dt=-4dx $\Rightarrow dx=\frac{-dt}{4}.$
 
Now substituting for x and dx we get,
 
$I=\int \sec^2t\big(\frac{-dt}{4}\big).$
 
$\;\;\;=\frac{-1}{4}\int\sec^2tdt$.
 
 
On integrating we get,
 
$\frac{-1}{4}\tan t+c.$
 
Substituting back for t we get,
 
$\int\sec^2(7-4x)=\frac{1}{4}\tan(7-4x)+c.$
 
 
 
 

 

answered Jan 29, 2013 by sreemathi.v
 
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