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

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

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integral of 5/(1+5x)

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

ln ∣ 1 + 5 x ∣ + C

Solution steps

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

Take the constant out:

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

= 5 \cdot \int \frac{1}{1 + 5 x} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 1 + 5 x

= 5 \cdot \frac{1}{5} ln ∣ 1 + 5 x ∣

Simplify

5 \cdot \frac{1}{5} ln ∣ 1 + 5 x ∣ : ln ∣ 1 + 5 x ∣

= ln ∣ 1 + 5 x ∣

Add a constant to the solution

= ln ∣ 1 + 5 x ∣ + C

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