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`