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

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

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

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

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

Solution steps

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

Multiply by the conjugate of

3 x^{2} - x : \frac{9 x ^{4} - x ^{2}}{3 x ^{2} + x}

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

Simplify

(\frac{9 x ^{4} - x ^{2}}{3 x ^{2} + x})^{2} : x^{2} (3 x - 1)^{2}

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

Apply Integration By Parts

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

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

\int 2 x (3 x^{3} - 3 x^{2} + x) d x = 2 (\frac{3 x ^{5}}{5} - \frac{3 x ^{4}}{4} + \frac{x ^{3}}{3})

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

Add a constant to the solution

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

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