What is the integral of 3x(3x-1)^2 ?

The integral of 3x(3x-1)^2 is 3((9x^4)/4-2x^3+(x^2)/2)+C

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integral of 3x(3x-1)^2

\int 3 x (3 x - 1)^{2} d x

3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2}) + C

Solution steps

\int 3 x (3 x - 1)^{2} d x

Take the constant out:

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

= 3 \cdot \int x (3 x - 1)^{2} d x

Expand

x (3 x - 1)^{2} : 9 x^{3} - 6 x^{2} + x

= 3 \cdot \int 9 x^{3} - 6 x^{2} + 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

= 3 (\int 9 x^{3} d x - \int 6 x^{2} d x + \int x d x)

\int 9 x^{3} d x = \frac{9 x ^{4}}{4}

\int 6 x^{2} d x = 2 x^{3}

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

= 3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2})

Add a constant to the solution

= 3 (\frac{9 x ^{4}}{4} - 2 x^{3} + \frac{x ^{2}}{2}) + C

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What is the integral of 3x(3x-1)^2 ?
The integral of 3x(3x-1)^2 is 3((9x^4)/4-2x^3+(x^2)/2)+C