How do you evaluate the integral #int x^3e^(x^2)"d"x#?

1 Answer
Harish Chandra Rajpoot
Jun 21, 2018

#\int x^3e^{x^2}\ dx=1/2(x^2-1)e^{x^2}+C#

Explanation:

Let #x^2=t\implies 2x=dt\ or \ xdx=\frac{dt}{2}#
#\therefore \int x^3e^{x^2}\ dx#
#=\int x^2e^{x^2}(xdx)#
#=\int te^t\frac{dt}{2}#
#=\frac{1}{2}\int te^t\ dt#
#=\1/2(t\int e^t \dt-\int (\frac{d}{dt}(t)\cdot \int e^t\ dt)dt)#
#=\1/2(te^t-\int (1\cdot e^t)dt)#
#=\1/2(te^t-\int e^tdt)#
#=\1/2(te^t-e^t)+C#
#=\1/2(t-1)e^t+C#
#=1/2(x^2-1)e^{x^2}+C#

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