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

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

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

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

\frac{e ^{4} π}{2 2} erfi (2 x) + C

Solution steps

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

Simplify

e^{2 x^{2} + 4} : e^{2 x^{2}} e^{4}

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

Take the constant out:

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

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

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

Apply u-substitution

= e^{4} \cdot \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

= e^{4} \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)

= e^{4} \frac{1}{2} \cdot \frac{π}{2} erfi (u)

Substitute back

u = 2 x

= e^{4} \frac{1}{2} \cdot \frac{π}{2} erfi (2 x)

Simplify

e^{4} \frac{1}{2} \cdot \frac{π}{2} erfi (2 x) : \frac{e ^{4} π}{2 2} erfi (2 x)

= \frac{e ^{4} π}{2 2} erfi (2 x)

Add a constant to the solution

= \frac{e ^{4} π}{2 2} erfi (2 x) + C

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