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

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

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

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

2 ln ∣ x + 1 ∣ + C

Solution steps

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

Simplify

\frac{x}{x ^{2} + x} : \frac{1}{x + 1}

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = x + 1

= 2 ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + C

\frac{s}{s + 4}

\frac{d y}{d x} = \frac{1}{x + y}

\frac{\partial}{\partial x} (5 x^{2} y - 5 x + 4 y^{3})

\int x^{2} (e^{x}) d x

x \to - 2 lim (\frac{2 x + 2}{x - 1})

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