Question

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

Answer 1

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

Explanation:

No, you simply need to observe that `x^4 = 1/5 d/dx x^5`, and your expression is thus of the form `e^f(x) * f'(x)`, constants apart.

The same idea can be expressed in terms of substitution: let `y=x^5`, then `dy = 5x^4 dx` (and thus `x^4 dx = dy/5`), and the integral becomes

`int e^{x^5} * x^4 dx -> int e^y 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`.