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

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

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

\int \frac{5}{( 5 x + 1 ) ln ( 5 x + 1 )} d x

ln ∣ ln (5 x + 1) ∣ + C

Solution steps

\int \frac{5}{( 5 x + 1 ) ln ( 5 x + 1 )} 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}{( 5 x + 1 ) ln ( 5 x + 1 )} d x

Apply u-substitution

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 5 \cdot \frac{1}{5} ln ∣ v ∣

Substitute back

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

Simplify

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

= ln ∣ ln (5 x + 1) ∣

Add a constant to the solution

= ln ∣ ln (5 x + 1) ∣ + C

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