What is the integral of (x-5)2x ?

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

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integral of (x-5)2x

\int (x - 5) 2 x d x

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

Solution steps

\int (x - 5) \cdot 2 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 - 5) x d x

Expand

(x - 5) x : x^{2} - 5 x

= 2 \cdot \int x^{2} - 5 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

= 2 (\int x^{2} d x - \int 5 x d x)

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

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

= 2 (\frac{x ^{3}}{3} - \frac{5 x ^{2}}{2})

Simplify

2 (\frac{x ^{3}}{3} - \frac{5 x ^{2}}{2}) : \frac{2 x ^{3}}{3} - 5 x^{2}

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

Add a constant to the solution

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

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