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

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

integral of 3x(x-2)

\int 3 x (x - 2) d x

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

Solution steps

\int 3 x (x - 2) d x

Take the constant out:

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

= 3 \cdot \int x (x - 2) d x

Expand

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

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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