What is the integral of x*(e^{x^2}-3)^2 ?

The integral of x*(e^{x^2}-3)^2 is 1/4 e^{2x^2}-3e^{x^2}+(9x^2)/2+C

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integral of x*(e^{x^2}-3)^2

\int x \cdot (e^{x^{2}} - 3)^{2} d x

\frac{1}{4} e^{2 x^{2}} - 3 e^{x^{2}} + \frac{9 x ^{2}}{2} + C

Solution steps

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

Expand

x (e^{x^{2}} - 3)^{2} : e^{2 x^{2}} x - 6 e^{x^{2}} x + 9 x

= \int e^{2 x^{2}} x - 6 e^{x^{2}} x + 9 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{2 x^{2}} x d x - \int 6 e^{x^{2}} x d x + \int 9 x d x

\int e^{2 x^{2}} x d x = \frac{1}{4} e^{2 x^{2}}

\int 6 e^{x^{2}} x d x = 3 e^{x^{2}}

\int 9 x d x = \frac{9 x ^{2}}{2}

= \frac{1}{4} e^{2 x^{2}} - 3 e^{x^{2}} + \frac{9 x ^{2}}{2}

Add a constant to the solution

= \frac{1}{4} e^{2 x^{2}} - 3 e^{x^{2}} + \frac{9 x ^{2}}{2} + C

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What is the integral of x*(e^{x^2}-3)^2 ?
The integral of x*(e^{x^2}-3)^2 is 1/4 e^{2x^2}-3e^{x^2}+(9x^2)/2+C