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

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

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integral of (x-2)(4x+2)

\int (x - 2) (4 x + 2) d x

\frac{4 x ^{3}}{3} - 3 x^{2} - 4 x + C

Solution steps

\int (x - 2) (4 x + 2) d x

(x - 2) (4 x + 2) = 2 (x - 2) (2 x + 1)

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

Take the constant out:

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

= 2 \cdot \int (x - 2) (2 x + 1) d x

Expand

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

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

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

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

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

\int 2 d x = 2 x

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

Simplify

2 (\frac{2 x ^{3}}{3} - \frac{3 x ^{2}}{2} - 2 x) : \frac{4 x ^{3}}{3} - 3 x^{2} - 4 x

= \frac{4 x ^{3}}{3} - 3 x^{2} - 4 x

Add a constant to the solution

= \frac{4 x ^{3}}{3} - 3 x^{2} - 4 x + C

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