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

The integral of (e^{2x^2}) is (sqrt(pi))/(2sqrt(2))\erfi(sqrt(2)x)+C

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

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

\frac{π}{2 2} erfi (2 x) + C

Solution steps

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

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

Apply u-substitution

: \int e^{u^{2}} \frac{1}{2} d u

= \int e^{u^{2}} \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{2} \cdot \frac{π}{2} erfi (u)

Substitute back

u = 2 x

= \frac{1}{2} \cdot \frac{π}{2} erfi (2 x)

Simplify

\frac{1}{2} \cdot \frac{π}{2} erfi (2 x) : \frac{π}{2 2} erfi (2 x)

= \frac{π}{2 2} erfi (2 x)

Add a constant to the solution

= \frac{π}{2 2} erfi (2 x) + C

\int \frac{sin ( ln ( x ^{3} ) )}{x} d x

a e^{3 x}

g (x) = (x + 2 x) e^{x}

\frac{1}{( s - 2 ) ^{2}}

\frac{d ^{3}}{d x ^{3}} (\frac{1}{x ^{2} + 1})

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