How do you integrate $int x^3 e^(x^2 ) dx$ using integration by parts?

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Answer 1 · 2016-09-01T19:50:50

$= 1/2e^(x^2) ( x^2 - 1 ) + C$

throughout , remember that $d/dx(e^(x^2) )= 2x e^(x^2)$

So:
$int x^3 e^(x^2 ) dx$

setting it up for IBP
$= int color(red)(x^2) d/dx(color(blue)(1/2 e^(x^2 ))) dx$

so by IBP
$= color(red)(x^2) (color(blue)(1/2 e^(x^2 )) ) - int d/dx( color(red)(x^2)) (color(blue)(1/2 e^(x^2 ))) dx$

$= 1/2 x^2 e^(x^2) - int 2 x * 1/2 e^(x^2 ) dx$

$= 1/2 x^2 e^(x^2) - int x e^(x^2 ) dx$

$= 1/2 x^2 e^(x^2) - int 1/2 d/dx( e^(x^2 )) dx$

$= 1/2 x^2 e^(x^2) - 1/2 e^(x^2 ) + C$

$= 1/2e^(x^2) ( x^2 - 1 ) + C$

How do you integrate int x^3 e^(x^2 ) dx int x^3 e^(x^2 ) dx using integration by parts?