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

The integral of 2^{2x^2+1} is (sqrt(pi))/(sqrt(2)\sqrt{ln(2))}\erfi(sqrt(ln(4))x)+C

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integral of 2^{2x^2+1}

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

\frac{π}{2 ln ( 2 )} erfi (ln (4) x) + C

Solution steps

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

2^{2 x^{2} + 1} = 2^{1} \cdot 2^{2 x^{2}}

= \int 2^{1} \cdot 2^{2 x^{2}} d x

Take the constant out:

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

= 2^{1} \cdot \int 2^{2 x^{2}} d x

Simplify

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

Simplify

2^{2 x^{2}} : 4^{x^{2}}

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

Use the nonelementary integral:

\int 4^{x^{2}} d x = \frac{π erfi ( x ln ( 4 ) )}{2 ln ( 4 )}

= 2 \cdot \frac{π erfi ( x ln ( 4 ) )}{2 ln ( 4 )}

Simplify

2 \cdot \frac{π erfi ( x ln ( 4 ) )}{2 ln ( 4 )} : \frac{π}{2 ln ( 2 )} erfi (ln (4) x)

= \frac{π}{2 ln ( 2 )} erfi (ln (4) x)

Add a constant to the solution

= \frac{π}{2 ln ( 2 )} erfi (ln (4) x) + C

What is the integral of 2^{2x^2+1} ?
The integral of 2^{2x^2+1} is (sqrt(pi))/(sqrt(2)\sqrt{ln(2))}\erfi(sqrt(ln(4))x)+C