What is the integral of (5^x)/(1+5^x) ?

The integral of (5^x)/(1+5^x) is 1/(ln(5))ln|1+5^x|+C

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integral of (5^x)/(1+5^x)

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

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

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{ln ( 5 )} ln ∣ u ∣

Substitute back

u = 1 + 5^{x}

= \frac{1}{ln ( 5 )} ln ∣ 1 + 5^{x} ∣

Add a constant to the solution

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

\frac{d}{d x} ((1000 + x)^{- \frac{1}{3}})

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f (x) = (x^{3} - 9 x)^{8}, (- 3, 0)

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