What is the integral of (x+5)/(2-x) ?

The integral of (x+5)/(2-x) is -x-7ln|-x+2|+C

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integral of (x+5)/(2-x)

\int \frac{x + 5}{2 - x} d x

- x - 7 ln ∣ - x + 2 ∣ + C

Solution steps

\int \frac{x + 5}{2 - x} d x

Expand

\frac{x + 5}{2 - x} : \frac{x}{2 - x} + \frac{5}{2 - x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{2 - x} d x + \int \frac{5}{2 - x} d x

\int \frac{x}{2 - x} d x = - 2 ln ∣ 2 - x ∣ - x

\int \frac{5}{2 - x} d x = - 5 ln ∣ 2 - x ∣

= - 2 ln ∣ 2 - x ∣ - x - 5 ln ∣ 2 - x ∣

Simplify

= - x - 7 ln ∣ - x + 2 ∣

Add a constant to the solution

= - x - 7 ln ∣ - x + 2 ∣ + C

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What is the integral of (x+5)/(2-x) ?
The integral of (x+5)/(2-x) is -x-7ln|-x+2|+C