Question

How to integrate xe^(5/4*x^4) dx?

Answer 1

\[xe ^{\frac{ 5 }{ 4 }x^4}dx\]

Answer 2

I guess integration by parts

Answer 3

@Kainui

Answer 4

\[\frac{\sqrt{5}}{2}x^2=u\]\[\sqrt{5}x dx = du\]

\[\int x e^{\frac{5}{4}x^4}dx = \frac{1}{\sqrt{5}} \int e^{u^2}du\]

Not looking too good unless we have some bounds

Answer 5

Power series solution is super easy though!

\[xe^{\frac{5}{4}x^4} = x \sum_{n=0}^\infty \frac{\left(\frac{5}{4}x^4\right)^n}{n!} = \sum_{n=0}^\infty \left(\frac{5}{4}\right)^n\frac{x^{4n+1}}{n!}\]

Now it's just a polynomial, which we can integrate:

\[\int \sum_{n=0}^\infty \left(\frac{5}{4}\right)^n\frac{x^{4n+1}}{n!} dx = \sum_{n=0}^\infty \int \left(\frac{5}{4}\right)^n\frac{x^{4n+1}}{n!}dx\]\[ = C+ \sum_{n=0}^\infty \left(\frac{5}{4}\right)^n \frac{1}{4n+2}\frac{x^{4n+2}}{n!}\]