What is the integral of ((2x-1))/((x-3)^2) ?

The integral of ((2x-1))/((x-3)^2) is 2ln|x-3|-5/(x-3)+C

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integral of ((2x-1))/((x-3)^2)

\int \frac{( 2 x - 1 )}{( x - 3 ) ^{2}} d x

2 ln ∣ x - 3 ∣ - \frac{5}{x - 3} + C

Solution steps

\int \frac{2 x - 1}{( x - 3 ) ^{2}} d x

Expand

\frac{2 x - 1}{( x - 3 ) ^{2}} : \frac{2 x}{( x - 3 ) ^{2}} - \frac{1}{( x - 3 ) ^{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{2 x}{( x - 3 ) ^{2}} d x - \int \frac{1}{( x - 3 ) ^{2}} d x

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

\int \frac{1}{( x - 3 ) ^{2}} d x = - \frac{1}{x - 3}

= 2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3})

Simplify

2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3}) : 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

= 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

Add a constant to the solution

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

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What is the integral of ((2x-1))/((x-3)^2) ?
The integral of ((2x-1))/((x-3)^2) is 2ln|x-3|-5/(x-3)+C