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

The integral of sqrt(x-1)-(x-1)^2 is 2/3 (x-1)^{3/2}-(x^3)/3+x^2-x+C

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integral of sqrt(x-1)-(x-1)^2

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

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

Solution steps

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of sqrt(x-1)-(x-1)^2 ?
The integral of sqrt(x-1)-(x-1)^2 is 2/3 (x-1)^{3/2}-(x^3)/3+x^2-x+C