What is the integral of 4x^2e^{3x^2} ?

The integral of 4x^2e^{3x^2} is 4(1/6 e^{3x^2}x-(sqrt(pi))/(12sqrt(3))\erfi(sqrt(3)x))+C

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integral of 4x^2e^{3x^2}

\int 4 x^{2} e^{3 x^{2}} d x

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

Solution steps

\int 4 x^{2} e^{3 x^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x^{2} e^{3 x^{2}} d x

Simplify

x^{2} : xx

= 4 \cdot \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

= 4 (\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)

= 4 (\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x))

Add a constant to the solution

= 4 (\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x)) + C

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What is the integral of 4x^2e^{3x^2} ?
The integral of 4x^2e^{3x^2} is 4(1/6 e^{3x^2}x-(sqrt(pi))/(12sqrt(3))\erfi(sqrt(3)x))+C