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

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

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

\int \frac{2 ( x + 1 )}{x ^{2} + 2 x} d x

ln x^{2} + 2 x + C

Solution steps

\int \frac{2 ( x + 1 )}{x ^{2} + 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 + 1}{x ^{2} + 2 x} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 2 x

= 2 \cdot \frac{1}{2} ln x^{2} + 2 x

Simplify

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

= ln x^{2} + 2 x

Add a constant to the solution

= ln x^{2} + 2 x + C

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