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

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

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integral of 1/(-1-x)

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

- ln ∣ 1 + x ∣ + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = - 1 - x

= - ln ∣ - 1 - x ∣

Apply absolute rule:

∣ - a ∣ = ∣ a ∣

∣ - 1 - x ∣ = ∣ 1 + x ∣

= - ln ∣ 1 + x ∣

Add a constant to the solution

= - ln ∣ 1 + x ∣ + C

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