What is the integral from 0 to L of x^2(L-x)(2L-x) ?

The integral from 0 to L of x^2(L-x)(2L-x) is -(3L^5)/4+(L^5)/5+(2L^5)/3

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integral from 0 to L of x^2(L-x)(2L-x)

\int_{0}^{L} x^{2} (L - x) (2 L - x) d x

- \frac{3 L ^{5}}{4} + \frac{L ^{5}}{5} + \frac{2 L ^{5}}{3}

Solution steps

\int_{0}^{L} x^{2} (L - x) (2 L - x) d x

Expand

(- x + 2 L) (L - x) x^{2} : - 3 L x^{3} + x^{4} + 2 L^{2} x^{2}

= \int_{0}^{L} - 3 L x^{3} + x^{4} + 2 L^{2} 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

= - \int_{0}^{L} 3 L x^{3} d x + \int_{0}^{L} x^{4} d x + \int_{0}^{L} 2 L^{2} x^{2} d x

\int_{0}^{L} 3 L x^{3} d x = \frac{3 L ^{5}}{4}

\int_{0}^{L} x^{4} d x = \frac{L ^{5}}{5}

\int_{0}^{L} 2 L^{2} x^{2} d x = \frac{2 L ^{5}}{3}

= - \frac{3 L ^{5}}{4} + \frac{L ^{5}}{5} + \frac{2 L ^{5}}{3}

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What is the integral from 0 to L of x^2(L-x)(2L-x) ?
The integral from 0 to L of x^2(L-x)(2L-x) is -(3L^5)/4+(L^5)/5+(2L^5)/3