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

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

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integral of-2/(1+x)

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

- 2 ln ∣ 1 + x ∣ + C

Solution steps

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

Apply u-substitution

= - 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 = 1 + x

= - 2 ln ∣ 1 + x ∣

Add a constant to the solution

= - 2 ln ∣ 1 + x ∣ + C