How do you integrate ∫x5e−x3?

1 Answer
Andrea S.
Nov 16, 2017

∫x5e−x3dx=−e−x3(x3+1)3+C

Explanation:

Substitute t=x3 so that dt=3x2dx and note that:

x5dx=x33⋅3x2dx=tdt3

We then have:

∫x5e−x3dx=13∫te−tdt

Now integrate by parts using t as integer factor:

∫te−tdt=−∫td(e−t)=−te−t+∫e−tdt=−e−t(t+1)+C

and undoing the substitution:

∫x5e−x3dx=−e−x3(x3+1)3+C

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