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

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

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

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

- \frac{2}{3} ln ∣ x ∣ + \frac{5}{3} ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int - \frac{2}{3 x} + \frac{5}{3 ( x + 3 )} d 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{2}{3 x} d x + \int \frac{5}{3 ( x + 3 )} d x

\int \frac{2}{3 x} d x = \frac{2}{3} ln ∣ x ∣

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

= - \frac{2}{3} ln ∣ x ∣ + \frac{5}{3} ln ∣ x + 3 ∣

Add a constant to the solution

= - \frac{2}{3} ln ∣ x ∣ + \frac{5}{3} ln ∣ x + 3 ∣ + C

n = 0 \sum \infty 2^{(3 - 2 n)}

x = 1 - y^{2}, x = y^{2} - 1

\frac{1 - t ^{2}}{( 1 + t ^{2} ) ^{2}}

x \to 2 lim ((2 x + 5)^{x - 3})

\frac{6 x}{x + 5}, at x = - 3

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