What is the integral of (2x^2-3x)/(x+1) ?

The integral of (2x^2-3x)/(x+1) is x^2-5x+5ln|x+1|-6+C

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integral of (2x^2-3x)/(x+1)

\int \frac{2 x ^{2} - 3 x}{x + 1} d x

x^{2} - 5 x + 5 ln ∣ x + 1 ∣ - 6 + C

Solution steps

\int \frac{2 x ^{2} - 3 x}{x + 1} d x

Expand

\frac{2 x ^{2} - 3 x}{x + 1} : \frac{2 x ^{2}}{x + 1} - \frac{3 x}{x + 1}

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{2 x ^{2}}{x + 1} d x - \int \frac{3 x}{x + 1} d x

\int \frac{2 x ^{2}}{x + 1} d x = x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3

\int \frac{3 x}{x + 1} d x = 3 (x + 1 - ln ∣ x + 1 ∣)

= x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3 - 3 (x + 1 - ln ∣ x + 1 ∣)

Simplify

x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3 - 3 (x + 1 - ln ∣ x + 1 ∣) : x^{2} - 5 x + 5 ln ∣ x + 1 ∣ - 6

= x^{2} - 5 x + 5 ln ∣ x + 1 ∣ - 6

Add a constant to the solution

= x^{2} - 5 x + 5 ln ∣ x + 1 ∣ - 6 + C

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What is the integral of (2x^2-3x)/(x+1) ?
The integral of (2x^2-3x)/(x+1) is x^2-5x+5ln|x+1|-6+C