What is the integral of (4x-5)/((x-2)^3) ?

The integral of (4x-5)/((x-2)^3) is 4(-1/(x-2)-1/((x-2)^2))+5/(2(x-2)^2)+C

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integral of (4x-5)/((x-2)^3)

\int \frac{4 x - 5}{( x - 2 ) ^{3}} d x

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

Solution steps

\int \frac{4 x - 5}{( x - 2 ) ^{3}} d x

Expand

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

= \int \frac{4 x}{( x - 2 ) ^{3}} - \frac{5}{( 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{4 x}{( x - 2 ) ^{3}} d x - \int \frac{5}{( x - 2 ) ^{3}} d x

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

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

= 4 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) - (- \frac{5}{2 ( x - 2 ) ^{2}})

Simplify

= 4 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + \frac{5}{2 ( x - 2 ) ^{2}}

Add a constant to the solution

= 4 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + \frac{5}{2 ( x - 2 ) ^{2}} + C

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What is the integral of (4x-5)/((x-2)^3) ?
The integral of (4x-5)/((x-2)^3) is 4(-1/(x-2)-1/((x-2)^2))+5/(2(x-2)^2)+C