What is the integral of (2x+3)/(2-x) ?

The integral of (2x+3)/(2-x) is 4-2x-7ln|2-x|+C

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integral of (2x+3)/(2-x)

\int \frac{2 x + 3}{2 - x} d x

4 - 2 x - 7 ln ∣ 2 - x ∣ + C

Solution steps

\int \frac{2 x + 3}{2 - x} d x

Apply u-substitution

: \int - \frac{- 2 u + 7}{u} d u

= \int - \frac{- 2 u + 7}{u} d u

Take the constant out:

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

= - \int \frac{- 2 u + 7}{u} d u

Expand

\frac{- 2 u + 7}{u} : - 2 + \frac{7}{u}

= - \int - 2 + \frac{7}{u} d u

Apply the Sum Rule:

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

= - (- \int 2 d u + \int \frac{7}{u} d u)

\int 2 d u = 2 u

\int \frac{7}{u} d u = 7 ln ∣ u ∣

= - (- 2 u + 7 ln ∣ u ∣)

Substitute back

u = 2 - x

= - (- 2 (2 - x) + 7 ln ∣ 2 - x ∣)

Simplify

- (- 2 (2 - x) + 7 ln ∣ 2 - x ∣) : 4 - 2 x - 7 ln ∣ 2 - x ∣

= 4 - 2 x - 7 ln ∣ 2 - x ∣

Add a constant to the solution

= 4 - 2 x - 7 ln ∣ 2 - x ∣ + C

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What is the integral of (2x+3)/(2-x) ?
The integral of (2x+3)/(2-x) is 4-2x-7ln|2-x|+C