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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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