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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{1}{5} ln ∣ 1 + 5 ln (x) ∣ + C

Solution steps

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

Apply u-substitution

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = 1 + 5 ln (x)

= \frac{1}{5} ln ∣ 1 + 5 ln (x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ 1 + 5 ln (x) ∣ + C

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