What is the intergration of the base of the natural log?

Question

what is the int (XE^(-X^2)

int( is the intergration of

^ is the what its powered by, so 2^2 is two squared

E is the base of the natural log

X ..of course the variable X

i know its intergration by parts, but please show it to me, i kept going in circle...

Answer 1

Actually, it's easier to use substitution.

Let u = - x^2, then du = -2x dx, and

∫ x * e^(- x^2) dx

= ( -1/2) ∫ e^u du

= [ ( -1/2) * e^u ] + C

= [ ( -1/2) * e^(- x^2) ] + C.

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Differentiate this result to check if you wish.

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Edit: Integration by parts won't work because however you try to divide up the function to use that technique, the e^(-x^2) factor will never reduce to something simpler.