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

The integral of (x-4)/((x-2)^3) is -1/(x-2)+1/((x-2)^2)+C

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

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

- \frac{1}{x - 2} + \frac{1}{( x - 2 ) ^{2}} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

= - \frac{1}{x - 2} + \frac{1}{( x - 2 ) ^{2}}

Add a constant to the solution

= - \frac{1}{x - 2} + \frac{1}{( x - 2 ) ^{2}} + C

What is the integral of (x-4)/((x-2)^3) ?
The integral of (x-4)/((x-2)^3) is -1/(x-2)+1/((x-2)^2)+C