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

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

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

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

- \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

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

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣

Add a constant to the solution

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

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