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

The integral of (x-3)/((x-5)x^2) is 2/25 ln|x-5|-2/25 ln|x|-3/(5x)+C

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

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

\frac{2}{25} ln ∣ x - 5 ∣ - \frac{2}{25} ln ∣ x ∣ - \frac{3}{5 x} + C

Solution steps

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

Take the partial fraction of

\frac{x - 3}{( x - 5 ) x ^{2}} : \frac{2}{25 ( x - 5 )} - \frac{2}{25 x} + \frac{3}{5 x ^{2}}

= \int \frac{2}{25 ( x - 5 )} - \frac{2}{25 x} + \frac{3}{5 x ^{2}} 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}{25 ( x - 5 )} d x - \int \frac{2}{25 x} d x + \int \frac{3}{5 x ^{2}} d x

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

\int \frac{2}{25 x} d x = \frac{2}{25} ln ∣ x ∣

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

= \frac{2}{25} ln ∣ x - 5 ∣ - \frac{2}{25} ln ∣ x ∣ - \frac{3}{5 x}

Add a constant to the solution

= \frac{2}{25} ln ∣ x - 5 ∣ - \frac{2}{25} ln ∣ x ∣ - \frac{3}{5 x} + C

(1.1) (3.4)

\int_{0}^{1} x^{3} x^{2} d x

\frac{\partial}{\partial x} (e^{- x^{4} - y^{3}})

\int_{1}^{1} x d x

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

What is the integral of (x-3)/((x-5)x^2) ?
The integral of (x-3)/((x-5)x^2) is 2/25 ln|x-5|-2/25 ln|x|-3/(5x)+C