What is the integral of (5^{3x})/(5^{3x)+7} ?

The integral of (5^{3x})/(5^{3x)+7} is 1/(3ln(5))ln|125^x+7|+C

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integral of (5^{3x})/(5^{3x)+7}

\int \frac{5 ^{3 x}}{5 ^{3 x} + 7} d x

\frac{1}{3 ln ( 5 )} ln ∣ 12 5^{x} + 7 ∣ + C

Solution steps

\int \frac{5 ^{3 x}}{5 ^{3 x} + 7} d x

Apply u-substitution

: \int \frac{1}{3 ln ( 5 ) u} d u

= \int \frac{1}{3 ln ( 5 ) u} d u

Take the constant out:

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

= \frac{1}{3 ln ( 5 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3 ln ( 5 )} ln ∣ u ∣

Substitute back

u = 5^{3 x} + 7

= \frac{1}{3 ln ( 5 )} ln 5^{3 x} + 7

5^{3 x} = 12 5^{x}

= \frac{1}{3 ln ( 5 )} ln ∣ 12 5^{x} + 7 ∣

Add a constant to the solution

= \frac{1}{3 ln ( 5 )} ln ∣ 12 5^{x} + 7 ∣ + C

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What is the integral of (5^{3x})/(5^{3x)+7} ?
The integral of (5^{3x})/(5^{3x)+7} is 1/(3ln(5))ln|125^x+7|+C