Ask Questions, Get Answers

Home  >>  CBSE XII  >>  Math  >>  Integrals

Integrate the function $\frac{\large x}{\large e^{x^2}}$

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 \frac{x}{e^{x^2}}dx $.
Put $x^2=t.$
$\;\;2xdx=dt \Rightarrow xdx=\frac{dt}{2}$.
Now substituting for x and dx we get,
$\int\frac{dt}{2e^t}=\frac{1}{2}\int e^{-t}dt.$
On integrating we get,
$\;\;\;\frac{1}{2}\int e^{-t}dt=\frac{-1}{2}e^{-t}+c.$
Substituting for t we get,
Hence $ \int\frac{x}{e^{x^2}}dx=\frac{-1}{2e^{x^2}}+c.$


answered Jan 29, 2013 by sreemathi.v