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

The integral of (2x^2-3)/(x^2+5x) is 2x-3/5 ln|x|-47/5 ln|x+5|+10+C

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

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

2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10 + C

Solution steps

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

Expand

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

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

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

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

Simplify

2 (x + 5 - 5 ln ∣ x + 5 ∣) - 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣) : 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10

= 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10

Add a constant to the solution

= 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10 + C

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What is the integral of (2x^2-3)/(x^2+5x) ?
The integral of (2x^2-3)/(x^2+5x) is 2x-3/5 ln|x|-47/5 ln|x+5|+10+C