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

1 Answer

  • (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,
$I=\int \sec^2t\big(\frac{-dt}{4}\big).$
On integrating we get,
$\frac{-1}{4}\tan t+c.$
Substituting back for t we get,


answered Jan 29, 2013 by sreemathi.v