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

The integral of x/((x+1)) is x+1-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 x/((x+1))

integral

\frac{x}{( x + 1 )}

x + 1 - ln ∣ x + 1 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

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

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 u - \int \frac{1}{u} d u

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = x + 1

= x + 1 - ln ∣ x + 1 ∣

Add a constant to the solution

= x + 1 - ln ∣ x + 1 ∣ + C

3 x^{3}

\frac{x}{( x ^{2} + y ^{2} )}

\frac{1}{( x ^{2} - a ^{2} )}

ln (x^{2} + 1)

\frac{1}{( a ^{2} + x ^{2} )}