Is integration by parts necessary to solve int x^4 e^(x^5)?

Oct 31, 2015

${e}^{{x}^{5}} / 5$.

Explanation:

No, you simply need to observe that ${x}^{4} = \frac{1}{5} \frac{d}{\mathrm{dx}} {x}^{5}$, and your expression is thus of the form ${e}^{f} \left(x\right) \cdot f ' \left(x\right)$, constants apart.

The same idea can be expressed in terms of substitution: let $y = {x}^{5}$, then $\mathrm{dy} = 5 {x}^{4} \mathrm{dx}$ (and thus ${x}^{4} \mathrm{dx} = \frac{\mathrm{dy}}{5}$), and the integral becomes

$\int {e}^{{x}^{5}} \cdot {x}^{4} \mathrm{dx} \to \int {e}^{y} \frac{\mathrm{dy}}{5}$

This integral is of course ${e}^{y} / 5$, since the exponential function equals its derivative and its integral. Substituting back $y = {x}^{5}$, you have the final result

${e}^{{x}^{5}} / 5$.