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

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

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integral of ax(x-4)

\int a x (x - 4) d x

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

Solution steps

\int a x (x - 4) d x

Take the constant out:

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

= a \cdot \int (- 4 + x) x d x

Expand

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

= a \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

= a (- \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}

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

Add a constant to the solution

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

\int x ln (2) d x

\int \frac{x ^{2}}{x ^{2} + x + 56} d x

f (x) = (sin (3 x))^{3 x}

\int e^{6 x} sin (e^{3 x}) d x

\frac{d}{d x} (2 x - x^{2})