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

1 Answer
Oct 31, 2015

#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#.