- #1

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The problem is in the title, but I'll rewrite it here simplified:

∫(x^2)exp(-x^2)dx

Subbing u = x^2 <=> x = sqrt(u) => dx = -0.5(u^(-1/2))du which becomes

-0.5∫(u/sqrt(u))exp(u)du = -0.5∫ sqrt(u)exp(u)du

So from here I tried integration by parts, u = sqrt(u), du = -0.5u^(-3/2)du,

dv = exp(u)du, v = exp(u)

uv - ∫vdu = sqrt(u)exp(u) + 0.5∫u^(-3/2)exp(u)du

From here I equatted this with -0.5∫sqrt(u)exp(u)du and noted

u^(-3/2) = (u^(-1/2))*(u^-1) = (u^(-1/2))/u

so then I did the algebra and got:

∫u^(-1/2)exp(u)du = u^(-1/2)(u/(u-1))exp(u) = sqrt(u)exp(u)/(u-1)

Am I on the right track? I am aware I can do this with the complementary error function or the gamma function but I'm trying to avoid using them. Any help is appreciated thank you.