What is the integral of (x-45)/(x^2(x+5)) ?

The integral of (x-45)/(x^2(x+5)) is 2ln|x|+9/x-2ln|x+5|+C

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integral of (x-45)/(x^2(x+5))

\int \frac{x - 45}{x ^{2} ( x + 5 )} d x

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

Solution steps

\int \frac{x - 45}{x ^{2} ( x + 5 )} d x

Take the partial fraction of

\frac{x - 45}{x ^{2} ( x + 5 )} : \frac{2}{x} - \frac{9}{x ^{2}} - \frac{2}{x + 5}

= \int \frac{2}{x} - \frac{9}{x ^{2}} - \frac{2}{x + 5} 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}{x} d x - \int \frac{9}{x ^{2}} d x - \int \frac{2}{x + 5} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{9}{x ^{2}} d x = - \frac{9}{x}

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x-45)/(x^2(x+5)) ?
The integral of (x-45)/(x^2(x+5)) is 2ln|x|+9/x-2ln|x+5|+C