What is the integral of (2x+13)/(x^2+3x-4) ?

The integral of (2x+13)/(x^2+3x-4) is 3ln|x-1|-ln|x+4|+C

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integral of (2x+13)/(x^2+3x-4)

\int \frac{2 x + 13}{x ^{2} + 3 x - 4} d x

3 ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{2 x + 13}{x ^{2} + 3 x - 4} d x

Expand

\frac{2 x + 13}{x ^{2} + 3 x - 4} : \frac{2 x}{x ^{2} + 3 x - 4} + \frac{13}{x ^{2} + 3 x - 4}

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 - 4} d x + \int \frac{13}{x ^{2} + 3 x - 4} d x

\int \frac{2 x}{x ^{2} + 3 x - 4} d x = 2 (\frac{1}{5} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x + 4 ∣)

\int \frac{13}{x ^{2} + 3 x - 4} d x = 13 (\frac{1}{5} ln ∣ x - 1 ∣ - \frac{1}{5} ln ∣ x + 4 ∣)

= 2 (\frac{1}{5} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x + 4 ∣) + 13 (\frac{1}{5} ln ∣ x - 1 ∣ - \frac{1}{5} ln ∣ x + 4 ∣)

Simplify

2 (\frac{1}{5} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x + 4 ∣) + 13 (\frac{1}{5} ln ∣ x - 1 ∣ - \frac{1}{5} ln ∣ x + 4 ∣) : 3 ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣

= 3 ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣

Add a constant to the solution

= 3 ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣ + C

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What is the integral of (2x+13)/(x^2+3x-4) ?
The integral of (2x+13)/(x^2+3x-4) is 3ln|x-1|-ln|x+4|+C