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

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

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integral of (2x-5)/(x(x-1)^3)

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

5 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + \frac{3}{2 ( x - 1 ) ^{2}} + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int \frac{5}{x - 1} d x = 5 ln ∣ x - 1 ∣

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

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

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

Simplify

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

Add a constant to the solution

= 5 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + \frac{3}{2 ( x - 1 ) ^{2}} + C

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