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Popular >

integral of x^2(x-3)^{10}

Solution

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

Solution

\frac{x ^{13}}{13} - \frac{5 x ^{12}}{2} + \frac{405 x ^{11}}{11} - 324 x^{10} + 1890 x^{9} - \frac{15309 x ^{8}}{2} + 21870 x^{7} - 43740 x^{6} + 59049 x^{5} - \frac{98415 x ^{4}}{2} + 19683 x^{3} + C

Solution steps

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

Expand

x^{2} (x - 3)^{10} : x^{12} - 30 x^{11} + 405 x^{10} - 3240 x^{9} + 17010 x^{8} - 61236 x^{7} + 153090 x^{6} - 262440 x^{5} + 295245 x^{4} - 196830 x^{3} + 59049 x^{2}

= \int x^{12} - 30 x^{11} + 405 x^{10} - 3240 x^{9} + 17010 x^{8} - 61236 x^{7} + 153090 x^{6} - 262440 x^{5} + 295245 x^{4} - 196830 x^{3} + 59049 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 x^{12} d x - \int 30 x^{11} d x + \int 405 x^{10} d x - \int 3240 x^{9} d x + \int 17010 x^{8} d x - \int 61236 x^{7} d x + \int 153090 x^{6} d x - \int 262440 x^{5} d x + \int 295245 x^{4} d x - \int 196830 x^{3} d x + \int 59049 x^{2} d x

\int x^{12} d x = \frac{x ^{13}}{13}

\int 30 x^{11} d x = \frac{5 x ^{12}}{2}

\int 405 x^{10} d x = \frac{405 x ^{11}}{11}

\int 3240 x^{9} d x = 324 x^{10}

\int 17010 x^{8} d x = 1890 x^{9}

\int 61236 x^{7} d x = \frac{15309 x ^{8}}{2}

\int 153090 x^{6} d x = 21870 x^{7}

\int 262440 x^{5} d x = 43740 x^{6}

\int 295245 x^{4} d x = 59049 x^{5}

\int 196830 x^{3} d x = \frac{98415 x ^{4}}{2}

\int 59049 x^{2} d x = 19683 x^{3}

= \frac{x ^{13}}{13} - \frac{5 x ^{12}}{2} + \frac{405 x ^{11}}{11} - 324 x^{10} + 1890 x^{9} - \frac{15309 x ^{8}}{2} + 21870 x^{7} - 43740 x^{6} + 59049 x^{5} - \frac{98415 x ^{4}}{2} + 19683 x^{3}

Add a constant to the solution

= \frac{x ^{13}}{13} - \frac{5 x ^{12}}{2} + \frac{405 x ^{11}}{11} - 324 x^{10} + 1890 x^{9} - \frac{15309 x ^{8}}{2} + 21870 x^{7} - 43740 x^{6} + 59049 x^{5} - \frac{98415 x ^{4}}{2} + 19683 x^{3} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^2(x-3)^{10} ?
The integral of x^2(x-3)^{10} is