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

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

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integral of (2x+1)(3-x)

\int (2 x + 1) (3 - x) d x

- \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 3 x + C

Solution steps

\int (2 x + 1) (3 - x) d x

Expand

(2 x + 1) (3 - x) : - 2 x^{2} + 5 x + 3

= \int - 2 x^{2} + 5 x + 3 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 5 x d x + \int 3 d x

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

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 3 d x = 3 x

= - \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 3 x

Add a constant to the solution

= - \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 3 x + C

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