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

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

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integral of 2/(2-x)

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

- 2 ln ∣ 2 - x ∣ + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- ln ∣ u ∣)

Substitute back

u = 2 - x

= 2 (- ln ∣ 2 - x ∣)

Simplify

= - 2 ln ∣ 2 - x ∣

Add a constant to the solution

= - 2 ln ∣ 2 - x ∣ + C