What is the integral of ((6x-8)/((x-3)^3)) ?

The integral of ((6x-8)/((x-3)^3)) is -6/(x-3)-5/((x-3)^2)+C

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integral of ((6x-8)/((x-3)^3))

\int (\frac{6 x - 8}{( x - 3 ) ^{3}}) d x

- \frac{6}{x - 3} - \frac{5}{( x - 3 ) ^{2}} + C

Solution steps

\int \frac{6 x - 8}{( x - 3 ) ^{3}} d x

Expand

\frac{6 x - 8}{( x - 3 ) ^{3}} : \frac{6 x}{( x - 3 ) ^{3}} - \frac{8}{( x - 3 ) ^{3}}

= \int \frac{6 x}{( x - 3 ) ^{3}} - \frac{8}{( x - 3 ) ^{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{6 x}{( x - 3 ) ^{3}} d x - \int \frac{8}{( x - 3 ) ^{3}} d x

\int \frac{6 x}{( x - 3 ) ^{3}} d x = 6 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}})

\int \frac{8}{( x - 3 ) ^{3}} d x = - \frac{4}{( x - 3 ) ^{2}}

= 6 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}) - (- \frac{4}{( x - 3 ) ^{2}})

Simplify

6 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}) - (- \frac{4}{( x - 3 ) ^{2}}) : - \frac{6}{x - 3} - \frac{5}{( x - 3 ) ^{2}}

= - \frac{6}{x - 3} - \frac{5}{( x - 3 ) ^{2}}

Add a constant to the solution

= - \frac{6}{x - 3} - \frac{5}{( x - 3 ) ^{2}} + C

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What is the integral of ((6x-8)/((x-3)^3)) ?
The integral of ((6x-8)/((x-3)^3)) is -6/(x-3)-5/((x-3)^2)+C