#### Question

# How to integrate x^5 e^x^3?

How do you integrate by parts x^5 e^x^3?

Thanks for any answers!

#### Answers

Use the change of variables t = x^3, dt = 3x2 dx first. Then use that x^5 = x^3 x2. Two x's are for the dt and 3 are the t.

∫ x^5 e^(x^3) dx = ∫ (1/3) t e^t dt

Now integrate by parts,

u = t, du = dt

dv = e^t dt , v = e^t

∫ (1/3) t e^t dt = (1/3)[ t e^t - ∫ e^t dt] = (1/3) (t e^t - e^t) + C

Substituting back gives the final solution

∫ x^5 e^(x^3) dx = (1/3)e^(x^3) [x^3 - 1] + C.

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