What is the integral of 2x(4-x) ?

The integral of 2x(4-x) is 2(2x^2-(x^3)/3)+C

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integral of 2x(4-x)

\int 2 x (4 - x) d x

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

Solution steps

\int 2 x (4 - x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (4 - x) d x

Expand

x (4 - x) : 4 x - x^{2}

= 2 \cdot \int 4 x - 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

= 2 (\int 4 x d x - \int x^{2} d x)

\int 4 x d x = 2 x^{2}

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

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

Add a constant to the solution

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