What is the integral of (x^2-5x+4)/(x+1) ?

The integral of (x^2-5x+4)/(x+1) is -7x+10ln|x+1|+1/2 (x+1)^2-7+C

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integral of (x^2-5x+4)/(x+1)

\int \frac{x ^{2} - 5 x + 4}{x + 1} d x

- 7 x + 10 ln ∣ x + 1 ∣ + \frac{1}{2} (x + 1)^{2} - 7 + C

Solution steps

\int \frac{x ^{2} - 5 x + 4}{x + 1} d x

Expand

\frac{x ^{2} - 5 x + 4}{x + 1} : \frac{x ^{2}}{x + 1} - \frac{5 x}{x + 1} + \frac{4}{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{x ^{2}}{x + 1} d x - \int \frac{5 x}{x + 1} d x + \int \frac{4}{x + 1} d x

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

\int \frac{5 x}{x + 1} d x = 5 (x + 1 - ln ∣ x + 1 ∣)

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= \frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣ - 5 (x + 1 - ln ∣ x + 1 ∣) + 4 ln ∣ x + 1 ∣

Simplify

\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣ - 5 (x + 1 - ln ∣ x + 1 ∣) + 4 ln ∣ x + 1 ∣ : - 7 x + 10 ln ∣ x + 1 ∣ + \frac{1}{2} (x + 1)^{2} - 7

= - 7 x + 10 ln ∣ x + 1 ∣ + \frac{1}{2} (x + 1)^{2} - 7

Add a constant to the solution

= - 7 x + 10 ln ∣ x + 1 ∣ + \frac{1}{2} (x + 1)^{2} - 7 + C

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What is the integral of (x^2-5x+4)/(x+1) ?
The integral of (x^2-5x+4)/(x+1) is -7x+10ln|x+1|+1/2 (x+1)^2-7+C