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Popular >

integral of x(x-1)^{10}

Solution

\int x (x - 1)^{10} d x

Solution

\frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2} + C

Solution steps

\int x (x - 1)^{10} d x

Expand

x (x - 1)^{10} : x^{11} - 10 x^{10} + 45 x^{9} - 120 x^{8} + 210 x^{7} - 252 x^{6} + 210 x^{5} - 120 x^{4} + 45 x^{3} - 10 x^{2} + x

= \int x^{11} - 10 x^{10} + 45 x^{9} - 120 x^{8} + 210 x^{7} - 252 x^{6} + 210 x^{5} - 120 x^{4} + 45 x^{3} - 10 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

= \int x^{11} d x - \int 10 x^{10} d x + \int 45 x^{9} d x - \int 120 x^{8} d x + \int 210 x^{7} d x - \int 252 x^{6} d x + \int 210 x^{5} d x - \int 120 x^{4} d x + \int 45 x^{3} d x - \int 10 x^{2} d x + \int x d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 10 x^{10} d x = \frac{10 x ^{11}}{11}

\int 45 x^{9} d x = \frac{9 x ^{10}}{2}

\int 120 x^{8} d x = \frac{40 x ^{9}}{3}

\int 210 x^{7} d x = \frac{105 x ^{8}}{4}

\int 252 x^{6} d x = 36 x^{7}

\int 210 x^{5} d x = 35 x^{6}

\int 120 x^{4} d x = 24 x^{5}

\int 45 x^{3} d x = \frac{45 x ^{4}}{4}

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

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

= \frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2} + C

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Frequently Asked Questions (FAQ)

What is the integral of x(x-1)^{10} ?
The integral of x(x-1)^{10} is