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

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

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

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

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

Solution steps

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

Expand

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

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

\int 2 d x = 2 x

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

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

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

Add a constant to the solution

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

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