What is the integral of 5^{8x+3} ?

The integral of 5^{8x+3} is (5^{8x+3})/(8ln(5))+C

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integral of 5^{8x+3}

\int 5^{8 x + 3} d x

\frac{5 ^{8 x + 3}}{8 ln ( 5 )} + C

Solution steps

\int 5^{8 x + 3} d x

Apply u-substitution

: \int 5^{u} \cdot \frac{1}{8} d u

= \int 5^{u} \cdot \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int 5^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{8} \cdot \frac{5 ^{u}}{ln ( 5 )}

Substitute back

u = 8 x + 3

= \frac{1}{8} \cdot \frac{5 ^{8 x + 3}}{ln ( 5 )}

Simplify

\frac{1}{8} \cdot \frac{5 ^{8 x + 3}}{ln ( 5 )} : \frac{5 ^{8 x + 3}}{8 ln ( 5 )}

= \frac{5 ^{8 x + 3}}{8 ln ( 5 )}

Add a constant to the solution

= \frac{5 ^{8 x + 3}}{8 ln ( 5 )} + C

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What is the integral of 5^{8x+3} ?
The integral of 5^{8x+3} is (5^{8x+3})/(8ln(5))+C