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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int \frac{2 x}{1 + x} d x

2 (1 + x - ln ∣ 1 + x ∣) + C

Solution steps

\int \frac{2 x}{1 + x} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{1 + x} d x

Apply u-substitution

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

= 2 \cdot \int \frac{u - 1}{u} d u

Expand

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

= 2 \cdot \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

= 2 (\int 1 d u - \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 (u - ln ∣ u ∣)

Substitute back

u = 1 + x

= 2 (1 + x - ln ∣ 1 + x ∣)

Add a constant to the solution

= 2 (1 + x - ln ∣ 1 + x ∣) + C

x \to 2 π lim (x (csc (x)))

\int_{0}^{1} (2 x^{2} + 1) d x

f (x) = (7 - x^{3})^{\frac{2}{3}}

y^{^{'}} - y = 2 x - 3

x, x - 6, x, 2

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