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

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

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

\int (x - 1) (2 - x) d x

\frac{3 x ^{2}}{2} - \frac{x ^{3}}{3} - 2 x + C

Solution steps

\int (x - 1) (2 - x) d x

Expand

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

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

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

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

\int 2 d x = 2 x

= \frac{3 x ^{2}}{2} - \frac{x ^{3}}{3} - 2 x

Add a constant to the solution

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

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