What is the integral of x(e^{x^2}+2) ?

The integral of x(e^{x^2}+2) is x^2+1/2 e^{x^2}+C

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integral of x(e^{x^2}+2)

\int x (e^{x^{2}} + 2) d x

x^{2} + \frac{1}{2} e^{x^{2}} + C

Solution steps

\int x (e^{x^{2}} + 2) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (e^{x^{2}} + 2) - \int e^{x^{2}} x^{3} d x

= \frac{1}{2} x^{2} (e^{x^{2}} + 2) - \int e^{x^{2}} x^{3} d x

\int e^{x^{2}} x^{3} d x = \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

= \frac{1}{2} x^{2} (e^{x^{2}} + 2) - \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

Simplify

\frac{1}{2} x^{2} (e^{x^{2}} + 2) - \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) : x^{2} + \frac{1}{2} e^{x^{2}}

= x^{2} + \frac{1}{2} e^{x^{2}}

Add a constant to the solution

= x^{2} + \frac{1}{2} e^{x^{2}} + C

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What is the integral of x(e^{x^2}+2) ?
The integral of x(e^{x^2}+2) is x^2+1/2 e^{x^2}+C