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

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

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integral of (2x^2+1)e^{(x^2)}

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

e^{x^{2}} x + C

Solution steps

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

Expand

(2 x^{2} + 1) e^{x^{2}} : 2 e^{x^{2}} x^{2} + e^{x^{2}}

= \int 2 e^{x^{2}} x^{2} + e^{x^{2}} 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 2 e^{x^{2}} x^{2} d x + \int e^{x^{2}} d x

\int 2 e^{x^{2}} x^{2} d x = e^{x^{2}} x - \frac{π}{2} erfi (x)

\int e^{x^{2}} d x = \frac{π}{2} erfi (x)

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

Add similar elements:

- \frac{π}{2} erfi (x) + \frac{π}{2} erfi (x) = 0

= e^{x^{2}} x

Add a constant to the solution

= e^{x^{2}} x + C

x \frac{d y}{d x} = \frac{1}{y ^{3}}

q^{^{''}} + 2 q^{^{'}} + \frac{1}{0.25} q = 50 cos (3 t)

\frac{\partial}{\partial y} (x arctan (x + 2 y))

e^{x} - e^{- x} - 2 x

\int 2 x (x^{2} + 1)^{- 6} d x

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