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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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