What is the integral of 5^{2x+1} ?

The integral of 5^{2x+1} is (5^{1+2x})/(2ln(5))+C

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integral of 5^{2x+1}

\int 5^{2 x + 1} d x

\frac{5 ^{1 + 2 x}}{2 ln ( 5 )} + C

Solution steps

\int 5^{2 x + 1} d x

Apply Integration By Parts

= 5^{2 x + 1} x - \int 2 ln (5) \cdot 5^{2 x + 1} x d x

\int 2 ln (5) \cdot 5^{2 x + 1} x d x = 5^{1 + 2 x} x - \frac{5 ^{1 + 2 x}}{2 ln ( 5 )}

= 5^{2 x + 1} x - (5^{1 + 2 x} x - \frac{5 ^{1 + 2 x}}{2 ln ( 5 )})

Simplify

= \frac{5 ^{1 + 2 x}}{2 ln ( 5 )}

Add a constant to the solution

= \frac{5 ^{1 + 2 x}}{2 ln ( 5 )} + C

What is the integral of 5^{2x+1} ?
The integral of 5^{2x+1} is (5^{1+2x})/(2ln(5))+C