What is the integral of (2x-4)/((x-3)(x-4)^2) ?

The integral of (2x-4)/((x-3)(x-4)^2) is 2ln|x-3|-2ln|x-4|-4/(x-4)+C

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integral of (2x-4)/((x-3)(x-4)^2)

\int \frac{2 x - 4}{( x - 3 ) ( x - 4 ) ^{2}} d x

2 ln ∣ x - 3 ∣ - 2 ln ∣ x - 4 ∣ - \frac{4}{x - 4} + C

Solution steps

\int \frac{2 x - 4}{( x - 3 ) ( x - 4 ) ^{2}} d x

Take the partial fraction of

\frac{2 x - 4}{( x - 3 ) ( x - 4 ) ^{2}} : \frac{2}{x - 3} - \frac{2}{x - 4} + \frac{4}{( x - 4 ) ^{2}}

= \int \frac{2}{x - 3} - \frac{2}{x - 4} + \frac{4}{( x - 4 ) ^{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}{x - 3} d x - \int \frac{2}{x - 4} d x + \int \frac{4}{( x - 4 ) ^{2}} d x

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

\int \frac{2}{x - 4} d x = 2 ln ∣ x - 4 ∣

\int \frac{4}{( x - 4 ) ^{2}} d x = - \frac{4}{x - 4}

= 2 ln ∣ x - 3 ∣ - 2 ln ∣ x - 4 ∣ - \frac{4}{x - 4}

Add a constant to the solution

= 2 ln ∣ x - 3 ∣ - 2 ln ∣ x - 4 ∣ - \frac{4}{x - 4} + C

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What is the integral of (2x-4)/((x-3)(x-4)^2) ?
The integral of (2x-4)/((x-3)(x-4)^2) is 2ln|x-3|-2ln|x-4|-4/(x-4)+C