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Integrate the function $e^{2x+3}$

1 Answer

  • 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 $\frac{dx}{dt}=g'(t).$
  • dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
Given $I=\int e^{2x+3}dx$.
Let 2x+3=t.
$\;\;2dx=dt \Rightarrow dx=\frac{dt}{2}$.
Now substituting for x and dx we get,
$\int e^{2x+3}dx=\frac{1}{2}\int e^t.dt.$
On integrating we get,
Substituting for t we get,
$\;\;\;\int e^{2x+3}dx=\frac{1}{2}e^{(2x+3)}+c.$
Hence $\int e^{2x+3}dx=\frac{1}{2}e^{(2x+3)}+c.$


answered Jan 28, 2013 by sreemathi.v