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

The integral of (x+2)/(x^2(x-5)) is -7/25 ln|x|+7/25 ln|x-5|+2/(5x)+C

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

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

- \frac{7}{25} ln ∣ x ∣ + \frac{7}{25} ln ∣ x - 5 ∣ + \frac{2}{5 x} + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x ^{2} ( x - 5 )} d x = 2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣)

= - \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣ + 2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣)

Simplify

- \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣ + 2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣) : - \frac{7}{25} ln ∣ x ∣ + \frac{7}{25} ln ∣ x - 5 ∣ + \frac{2}{5 x}

= - \frac{7}{25} ln ∣ x ∣ + \frac{7}{25} ln ∣ x - 5 ∣ + \frac{2}{5 x}

Add a constant to the solution

= - \frac{7}{25} ln ∣ x ∣ + \frac{7}{25} ln ∣ x - 5 ∣ + \frac{2}{5 x} + C

What is the integral of (x+2)/(x^2(x-5)) ?
The integral of (x+2)/(x^2(x-5)) is -7/25 ln|x|+7/25 ln|x-5|+2/(5x)+C