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

1 Answer
KillerBunny
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.

Related questions
See all questions in Integration by Parts
Impact of this question
1408 views around the world
You can reuse this answer
Creative Commons License