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

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

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

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

- \frac{1}{x - 5} - \frac{1}{( x - 5 ) ^{2}} + C

Solution steps

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

Expand

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

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

\int \frac{x}{( x - 5 ) ^{3}} d x = - \frac{1}{x - 5} - \frac{5}{2 ( x - 5 ) ^{2}}

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

= - \frac{1}{x - 5} - \frac{5}{2 ( x - 5 ) ^{2}} - (- \frac{3}{2 ( x - 5 ) ^{2}})

Simplify

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

= - \frac{1}{x - 5} - \frac{1}{( x - 5 ) ^{2}}

Add a constant to the solution

= - \frac{1}{x - 5} - \frac{1}{( x - 5 ) ^{2}} + C

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