What is the integral of 1/(x(5+2ln(x))) ?

The integral of 1/(x(5+2ln(x))) is 1/2 ln|5+2ln(x)|+C

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integral of 1/(x(5+2ln(x)))

\int \frac{1}{x ( 5 + 2 ln ( x ) )} d x

\frac{1}{2} ln ∣ 5 + 2 ln (x) ∣ + C

Solution steps

\int \frac{1}{x ( 5 + 2 ln ( x ) )} d x

Expand

x (5 + 2 ln (x)) : 5 x + 2 x ln (x)

= \int \frac{1}{5 x + 2 x ln ( x )} d x

Apply u-substitution

= \int \frac{1}{5 + 2 u} d u

Apply u-substitution

= \int \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln ∣ 5 + 2 ln (x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 5 + 2 ln (x) ∣ + C

What is the integral of 1/(x(5+2ln(x))) ?
The integral of 1/(x(5+2ln(x))) is 1/2 ln|5+2ln(x)|+C