What is the integral of 2x^2(x-2)(4x-5) ?

The integral of 2x^2(x-2)(4x-5) is 2((4x^5)/5-(13x^4)/4+(10x^3)/3)+C

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integral of 2x^2(x-2)(4x-5)

\int 2 x^{2} (x - 2) (4 x - 5) d x

2 (\frac{4 x ^{5}}{5} - \frac{13 x ^{4}}{4} + \frac{10 x ^{3}}{3}) + C

Solution steps

\int 2 x^{2} (x - 2) (4 x - 5) 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} (x - 2) (4 x - 5) d x

Expand

x^{2} (x - 2) (4 x - 5) : 4 x^{4} - 13 x^{3} + 10 x^{2}

= 2 \cdot \int 4 x^{4} - 13 x^{3} + 10 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^{4} d x - \int 13 x^{3} d x + \int 10 x^{2} d x)

\int 4 x^{4} d x = \frac{4 x ^{5}}{5}

\int 13 x^{3} d x = \frac{13 x ^{4}}{4}

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

= 2 (\frac{4 x ^{5}}{5} - \frac{13 x ^{4}}{4} + \frac{10 x ^{3}}{3})

Add a constant to the solution

= 2 (\frac{4 x ^{5}}{5} - \frac{13 x ^{4}}{4} + \frac{10 x ^{3}}{3}) + C

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What is the integral of 2x^2(x-2)(4x-5) ?
The integral of 2x^2(x-2)(4x-5) is 2((4x^5)/5-(13x^4)/4+(10x^3)/3)+C