What is the integral of (2)5^{x^2+7} ?

The integral of (2)5^{x^2+7} is (78125sqrt(pi))/(sqrt(ln(5)))\erfi(sqrt(ln(5))x)+C

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integral of (2)5^{x^2+7}

\int (2) 5^{x^{2} + 7} d x

\frac{78125 π}{ln ( 5 )} erfi (ln (5) x) + C

Solution steps

\int 2 \cdot 5^{x^{2} + 7} d x

Take the constant out:

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

= 2 \cdot \int 5^{x^{2} + 7} d x

5^{x^{2} + 7} = 5^{7} \cdot 5^{x^{2}}

= 2 \cdot \int 5^{7} \cdot 5^{x^{2}} d x

Take the constant out:

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

= 2 \cdot 5^{7} \cdot \int 5^{x^{2}} d x

Simplify

= 2 \cdot 78125 \cdot \int 5^{x^{2}} d x

Use the nonelementary integral:

\int 5^{x^{2}} d x = \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )}

= 2 \cdot 78125 \cdot \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )}

Simplify

2 \cdot 78125 \cdot \frac{π erfi ( x ln ( 5 ) )}{2 ln ( 5 )} : \frac{78125 π}{ln ( 5 )} erfi (ln (5) x)

= \frac{78125 π}{ln ( 5 )} erfi (ln (5) x)

Add a constant to the solution

= \frac{78125 π}{ln ( 5 )} erfi (ln (5) x) + C

What is the integral of (2)5^{x^2+7} ?
The integral of (2)5^{x^2+7} is (78125sqrt(pi))/(sqrt(ln(5)))\erfi(sqrt(ln(5))x)+C