What is the integral of (7x^2+8x-2)/(x^2+2x) ?

The integral of (7x^2+8x-2)/(x^2+2x) is 7x+14-5ln|x+2|-ln|x|+C

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integral of (7x^2+8x-2)/(x^2+2x)

\int \frac{7 x ^{2} + 8 x - 2}{x ^{2} + 2 x} d x

7 x + 14 - 5 ln ∣ x + 2 ∣ - ln ∣ x ∣ + C

Solution steps

\int \frac{7 x ^{2} + 8 x - 2}{x ^{2} + 2 x} d x

Expand

\frac{7 x ^{2} + 8 x - 2}{x ^{2} + 2 x} : \frac{7 x ^{2}}{x ^{2} + 2 x} + \frac{8 x}{x ^{2} + 2 x} - \frac{2}{x ^{2} + 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{7 x ^{2}}{x ^{2} + 2 x} d x + \int \frac{8 x}{x ^{2} + 2 x} d x - \int \frac{2}{x ^{2} + 2 x} d x

\int \frac{7 x ^{2}}{x ^{2} + 2 x} d x = 7 (x + 2 - 2 ln ∣ x + 2 ∣)

\int \frac{8 x}{x ^{2} + 2 x} d x = 8 ln ∣ x + 2 ∣

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

= 7 (x + 2 - 2 ln ∣ x + 2 ∣) + 8 ln ∣ x + 2 ∣ - (- ln ∣ x + 2 ∣ + ln ∣ x ∣)

Simplify

7 (x + 2 - 2 ln ∣ x + 2 ∣) + 8 ln ∣ x + 2 ∣ - (- ln ∣ x + 2 ∣ + ln ∣ x ∣) : 7 x + 14 - 5 ln ∣ x + 2 ∣ - ln ∣ x ∣

= 7 x + 14 - 5 ln ∣ x + 2 ∣ - ln ∣ x ∣

Add a constant to the solution

= 7 x + 14 - 5 ln ∣ x + 2 ∣ - ln ∣ x ∣ + C

What is the integral of (7x^2+8x-2)/(x^2+2x) ?
The integral of (7x^2+8x-2)/(x^2+2x) is 7x+14-5ln|x+2|-ln|x|+C