What is the integral of 4/(-2x+2) ?

The integral of 4/(-2x+2) is -2ln|-2x+2|+C

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integral of 4/(-2x+2)

\int \frac{4}{- 2 x + 2} d x

- 2 ln ∣ - 2 x + 2 ∣ + C

Solution steps

\int \frac{4}{- 2 x + 2} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{- 2 x + 2} d x

Apply u-substitution

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

= 4 \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

= 4 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- \frac{1}{2} ln ∣ u ∣)

Substitute back

u = - 2 x + 2

= 4 (- \frac{1}{2} ln ∣ - 2 x + 2 ∣)

Simplify

4 (- \frac{1}{2} ln ∣ - 2 x + 2 ∣) : - 2 ln ∣ - 2 x + 2 ∣

= - 2 ln ∣ - 2 x + 2 ∣

Add a constant to the solution

= - 2 ln ∣ - 2 x + 2 ∣ + C

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