What is the integral of x^2e^{3x^2} ?

The integral of x^2e^{3x^2} is 1/6 e^{3x^2}x-(sqrt(pi))/(12sqrt(3))\erfi(sqrt(3)x)+C

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integral of x^2e^{3x^2}

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

\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x) + C

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

: \frac{1}{6} e^{3 x^{2}} x - \int \frac{1}{6} e^{3 x^{2}} d x

= \frac{1}{6} e^{3 x^{2}} x - \int \frac{1}{6} e^{3 x^{2}} d x

\int \frac{1}{6} e^{3 x^{2}} d x = \frac{π}{12 3} erfi (3 x)

= \frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x)

Add a constant to the solution

= \frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x) + C

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

\int \frac{1}{x ^{2}} 5 + \frac{1}{x} d x

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\int \frac{1}{u ^{2} - 2 u + 1} d u

What is the integral of x^2e^{3x^2} ?
The integral of x^2e^{3x^2} is 1/6 e^{3x^2}x-(sqrt(pi))/(12sqrt(3))\erfi(sqrt(3)x)+C