Browse Questions

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

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 $\frac{dx}{dt}=g'(t).$
• 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,

$\;\;\;\frac{1}{2}\int e^{-t}dt=\frac{-1}{2}e^{-t}+c.$

$\;\;\;\quad=\frac{-1}{2e^t}+c.$

Substituting for t we get,

$\;\;\;\quad=\frac{-1}{2e^{x^2}}+c.$

Hence $\int\frac{x}{e^{x^2}}dx=\frac{-1}{2e^{x^2}}+c.$