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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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