What is the integral of (2x+5)/(x^2+3x-10) ?

The integral of (2x+5)/(x^2+3x-10) is 9/7 ln|x-2|+5/7 ln|x+5|+C

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integral of (2x+5)/(x^2+3x-10)

\int \frac{2 x + 5}{x ^{2} + 3 x - 10} d x

\frac{9}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{2 x + 5}{x ^{2} + 3 x - 10} d x

Expand

\frac{2 x + 5}{x ^{2} + 3 x - 10} : \frac{2 x}{x ^{2} + 3 x - 10} + \frac{5}{x ^{2} + 3 x - 10}

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 x}{x ^{2} + 3 x - 10} d x + \int \frac{5}{x ^{2} + 3 x - 10} d x

\int \frac{2 x}{x ^{2} + 3 x - 10} d x = 2 (\frac{2}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣)

\int \frac{5}{x ^{2} + 3 x - 10} d x = 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣)

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

Simplify

2 (\frac{2}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣) + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣) : \frac{9}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣

= \frac{9}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣

Add a constant to the solution

= \frac{9}{7} ln ∣ x - 2 ∣ + \frac{5}{7} ln ∣ x + 5 ∣ + C

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What is the integral of (2x+5)/(x^2+3x-10) ?
The integral of (2x+5)/(x^2+3x-10) is 9/7 ln|x-2|+5/7 ln|x+5|+C