What is the integral of (x^2-x+5)/(x+3) ?

The integral of (x^2-x+5)/(x+3) is -7x+17ln|x+3|+1/2 (x+3)^2-21+C

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integral of (x^2-x+5)/(x+3)

\int \frac{x ^{2} - x + 5}{x + 3} d x

- 7 x + 17 ln ∣ x + 3 ∣ + \frac{1}{2} (x + 3)^{2} - 21 + C

Solution steps

\int \frac{x ^{2} - x + 5}{x + 3} d x

Expand

\frac{x ^{2} - x + 5}{x + 3} : \frac{x ^{2}}{x + 3} - \frac{x}{x + 3} + \frac{5}{x + 3}

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 + 3} d x - \int \frac{x}{x + 3} d x + \int \frac{5}{x + 3} d x

\int \frac{x ^{2}}{x + 3} d x = \frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣

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

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

= \frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - (x + 3 - 3 ln ∣ x + 3 ∣) + 5 ln ∣ x + 3 ∣

Simplify

\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - (x + 3 - 3 ln ∣ x + 3 ∣) + 5 ln ∣ x + 3 ∣ : - 7 x + 17 ln ∣ x + 3 ∣ + \frac{1}{2} (x + 3)^{2} - 21

= - 7 x + 17 ln ∣ x + 3 ∣ + \frac{1}{2} (x + 3)^{2} - 21

Add a constant to the solution

= - 7 x + 17 ln ∣ x + 3 ∣ + \frac{1}{2} (x + 3)^{2} - 21 + C

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

\int (3 x^{2} + 4 x - 5) d x

\int \frac{2 x - 5}{x ^{2} - 5 x + 3} d x

\int 12 x - 3 d x

\int (x + 2) d x

What is the integral of (x^2-x+5)/(x+3) ?
The integral of (x^2-x+5)/(x+3) is -7x+17ln|x+3|+1/2 (x+3)^2-21+C