What is the integral of (x^4)/((x-1)^3) ?

The integral of (x^4)/((x-1)^3) is -(x^4)/(2(x-1)^2)+x^2+4x-2/(x-1)+6ln|x-1|-5+C

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integral of (x^4)/((x-1)^3)

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

- \frac{x ^{4}}{2 ( x - 1 ) ^{2}} + x^{2} + 4 x - \frac{2}{x - 1} + 6 ln ∣ x - 1 ∣ - 5 + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2 x ^{3}}{( x - 1 ) ^{2}} d x = - x^{2} - 4 x + \frac{2}{x - 1} - 6 ln ∣ x - 1 ∣ + 5

= - \frac{x ^{4}}{2 ( x - 1 ) ^{2}} - (- x^{2} - 4 x + \frac{2}{x - 1} - 6 ln ∣ x - 1 ∣ + 5)

Simplify

= - \frac{x ^{4}}{2 ( x - 1 ) ^{2}} + x^{2} + 4 x - \frac{2}{x - 1} + 6 ln ∣ x - 1 ∣ - 5

Add a constant to the solution

= - \frac{x ^{4}}{2 ( x - 1 ) ^{2}} + x^{2} + 4 x - \frac{2}{x - 1} + 6 ln ∣ x - 1 ∣ - 5 + C

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What is the integral of (x^4)/((x-1)^3) ?
The integral of (x^4)/((x-1)^3) is -(x^4)/(2(x-1)^2)+x^2+4x-2/(x-1)+6ln|x-1|-5+C