What is the integral of x^5*e^(x^2) Using u substituition and integration by parts.?

Question

Alright, well, I've gotten far enough to know that I'm missing something key. I've used integration by parts 4 times in a row with the resulting answer:

(2x^6e^x^2)-(20x^6e^x^2)+
(200x^6e^x^2)-
*integral*1000x^5e^x^2

I know you're supposed to factor something and divide by something...but I missed lecture today!

Anyone have an answer? ASAP if possible.

Thanks guys!

? · Accepted Answer

Let u = x^2; du = 2x * dx

∫x^5*e^(x^2) dx
= (1/2) * ∫u^2 * e^u du
Use integration by parts, U = u^2, dU = 2u, dV = e^u, V = e^u :
= (1/2) * [u^2e^u - 2∫ue^u du]
Integration by parts again, U = u, dU = 1, dV = d^u, V = e^u :
= (1/2) * [u^2e^u - 2[ue^u - ∫e^u du] ]
= (1/2) * [u^2e^u - 2[ue^u - e^u] ]
= (1/2) * [u^2e^u - 2ue^u + 2e^u  ]
= (1/2) * [x^4e^(x^2) - 2x^2e^(x^2) + 2e^(x^2)  ]
= e^(x^2) * (x^4 - 2x^2 + 2) / 2

Seto · Answer

u = e^(x^2) --> du = 2xe(x^2)
dv = x^5 --> v = (x^6)/6

--> = e(x^2)*(x^4 - 2x^2 +2)/2

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What is the integral of x^5*e^(x^2) Using u substituition and integration by parts.?

2 Answers