What is the integral of 1/(x(5+3ln(x))) ?

The integral of 1/(x(5+3ln(x))) is 1/3 ln|5+3ln(x)|+C

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integral of 1/(x(5+3ln(x)))

\int \frac{1}{x ( 5 + 3 ln ( x ) )} d x

\frac{1}{3} ln ∣ 5 + 3 ln (x) ∣ + C

Solution steps

\int \frac{1}{x ( 5 + 3 ln ( x ) )} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 5 + 3 ln (x)

= \frac{1}{3} ln ∣ 5 + 3 ln (x) ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ 5 + 3 ln (x) ∣ + C

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What is the integral of 1/(x(5+3ln(x))) ?
The integral of 1/(x(5+3ln(x))) is 1/3 ln|5+3ln(x)|+C