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

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

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

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

ln ∣ x - 5 ∣ - \frac{2}{x - 5} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

= ln ∣ x - 5 ∣ - \frac{2}{x - 5}

Add a constant to the solution

= ln ∣ x - 5 ∣ - \frac{2}{x - 5} + C

\int_{0}^{x^{2}} e^{- t^{2}} d t

\int_{0}^{3} x^{3} x^{2} + 9 d x

\frac{1}{2} e^{x}

\int sin (t) e^{- s t} d t

\frac{\partial}{\partial x} (e^{x y z^{9}})

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