Question

Use a change of variables to find the indefinite integral xe^(x^2) dx

Answer 1

$\displaystyle\int xe^{x^2}\,\mathrm dx=\frac12\int2xe^{x^2}\,\mathrm dx$

If we take $y=x^2$, then $\mathrm dy=2x\,\mathrm dx$, and we can write the above integral as


$\displaystyle\frac12\int e^y\,\mathrm dy=\frac12e^y+C$

and undoing the substitution, we end up with

$\displaystyle\int xe^{x^2}\,\mathrm dx=\frac12e^{x^2}+C$