What is the integral of 2x(3x-10)^4 ?

The integral of 2x(3x-10)^4 is 27x^6-432x^5+2700x^4-8000x^3+10000x^2+C

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integral of 2x(3x-10)^4

\int 2 x (3 x - 10)^{4} d x

27 x^{6} - 432 x^{5} + 2700 x^{4} - 8000 x^{3} + 10000 x^{2} + C

Solution steps

\int 2 x (3 x - 10)^{4} d x

Take the constant out:

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

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

Expand

x (3 x - 10)^{4} : 81 x^{5} - 1080 x^{4} + 5400 x^{3} - 12000 x^{2} + 10000 x

= 2 \cdot \int 81 x^{5} - 1080 x^{4} + 5400 x^{3} - 12000 x^{2} + 10000 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 81 x^{5} d x - \int 1080 x^{4} d x + \int 5400 x^{3} d x - \int 12000 x^{2} d x + \int 10000 x d x)

\int 81 x^{5} d x = \frac{27 x ^{6}}{2}

\int 1080 x^{4} d x = 216 x^{5}

\int 5400 x^{3} d x = 1350 x^{4}

\int 12000 x^{2} d x = 4000 x^{3}

\int 10000 x d x = 5000 x^{2}

= 2 (\frac{27 x ^{6}}{2} - 216 x^{5} + 1350 x^{4} - 4000 x^{3} + 5000 x^{2})

Simplify

2 (\frac{27 x ^{6}}{2} - 216 x^{5} + 1350 x^{4} - 4000 x^{3} + 5000 x^{2}) : 27 x^{6} - 432 x^{5} + 2700 x^{4} - 8000 x^{3} + 10000 x^{2}

= 27 x^{6} - 432 x^{5} + 2700 x^{4} - 8000 x^{3} + 10000 x^{2}

Add a constant to the solution

= 27 x^{6} - 432 x^{5} + 2700 x^{4} - 8000 x^{3} + 10000 x^{2} + C

2 x - 2 y

\frac{\partial}{\partial y} (e^{- x} sin (yz))

\int 7 - 6 x - x^{2} d x

\frac{d}{d x} (\frac{x ^{2}}{( x ^{2} - 1 )})

f (x) = 4 - 3 ln (x), at x = 1

What is the integral of 2x(3x-10)^4 ?
The integral of 2x(3x-10)^4 is 27x^6-432x^5+2700x^4-8000x^3+10000x^2+C