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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

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

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

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= v - ln ∣ v ∣

Substitute back

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣

Add a constant to the solution

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

\int (cos (x))^{2} (sin (x))^{2} d x

\int \frac{cos ( x )}{9 + sin ( x )} d x

\int \frac{- 8}{x x + 1} d x

\int (cos (π) x) d x

\int \frac{4}{9 + 2 x ^{2}} d x

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