What is the integral of (x-15)/(x^2(x+3)) ?

The integral of (x-15)/(x^2(x+3)) is 2ln|x|+5/x-2ln|x+3|+C

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integral of (x-15)/(x^2(x+3))

\int \frac{x - 15}{x ^{2} ( x + 3 )} d x

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

Solution steps

\int \frac{x - 15}{x ^{2} ( x + 3 )} d x

Take the partial fraction of

\frac{x - 15}{x ^{2} ( x + 3 )} : \frac{2}{x} - \frac{5}{x ^{2}} - \frac{2}{x + 3}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

Simplify

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

Add a constant to the solution

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

2 y^{^{''}} - 16 y^{^{'}} + 32 y = 4 e^{4 t}

x \to 2 lim (x^{2} - 4 x - 2)

4 x^{2} + 3 x

\frac{\partial}{\partial x} (\frac{108}{x})

\int \frac{1}{e ^{3 t} + e ^{t}} d t

What is the integral of (x-15)/(x^2(x+3)) ?
The integral of (x-15)/(x^2(x+3)) is 2ln|x|+5/x-2ln|x+3|+C