What is the integral of 5/((5x+3)ln(5x+3)) ?

The integral of 5/((5x+3)ln(5x+3)) is ln|ln(5x+3)|+C

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integral of 5/((5x+3)ln(5x+3))

\int \frac{5}{( 5 x + 3 ) ln ( 5 x + 3 )} d x

ln ∣ ln (5 x + 3) ∣ + C

Solution steps

\int \frac{5}{( 5 x + 3 ) ln ( 5 x + 3 )} 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 + 3 ) ln ( 5 x + 3 )} d x

Apply u-substitution

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

= 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

: \int \frac{1}{v} d v

= 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 + 3) ∣

Simplify

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

= ln ∣ ln (5 x + 3) ∣

Add a constant to the solution

= ln ∣ ln (5 x + 3) ∣ + C

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What is the integral of 5/((5x+3)ln(5x+3)) ?
The integral of 5/((5x+3)ln(5x+3)) is ln|ln(5x+3)|+C