Using U – Substitution or Integration by Parts, evaluate the Indefinite Integral of the following. ∫x^4e(^-x^3)dx

Question

Using U – Substitution or Integration by Parts, evaluate the Indefinite

Integral of the following.

∫ x⁴e^{-x^3}dx

Answer 1

I(x)=∫ x⁴e^{-x^3}dx

let u=-x²/3,,,,,,,,,,,,,,,,,dv=-3x²e^{-x^3}dx

du=-(2/3)xdx,,,,,,,,,,,,,,,v=e^{-x^3}

I(x)=-(1/3)x²e^{-x^3}+ ∫ (2/3)xe^{-x^3}dx

that is as far as integration by parts will go

in another possibility is saying x⁴e^{-x^3} = sum of x⁴(1-x^-3+x^-6/2!-x^-9/3! +/-...)

and integrate.

Answer 2

integral of x^4(e^-x^3)dx

=- the gamma function (5/3, x^3)/3 + C

u substitution u=x^5