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Popular >

integral of x(3x-1)^{10}

Solution

\int x (3 x - 1)^{10} d x

Solution

\frac{19683 x ^{12}}{4} - \frac{196830 x ^{11}}{11} + \frac{59049 x ^{10}}{2} - 29160 x^{9} + \frac{76545 x ^{8}}{4} - 8748 x^{7} + 2835 x^{6} - 648 x^{5} + \frac{405 x ^{4}}{4} - 10 x^{3} + \frac{x ^{2}}{2} + C

Solution steps

\int x (3 x - 1)^{10} d x

Expand

x (3 x - 1)^{10} : 59049 x^{11} - 196830 x^{10} + 295245 x^{9} - 262440 x^{8} + 153090 x^{7} - 61236 x^{6} + 17010 x^{5} - 3240 x^{4} + 405 x^{3} - 30 x^{2} + x

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

\int 59049 x^{11} d x = \frac{19683 x ^{12}}{4}

\int 196830 x^{10} d x = \frac{196830 x ^{11}}{11}

\int 295245 x^{9} d x = \frac{59049 x ^{10}}{2}

\int 262440 x^{8} d x = 29160 x^{9}

\int 153090 x^{7} d x = \frac{76545 x ^{8}}{4}

\int 61236 x^{6} d x = 8748 x^{7}

\int 17010 x^{5} d x = 2835 x^{6}

\int 3240 x^{4} d x = 648 x^{5}

\int 405 x^{3} d x = \frac{405 x ^{4}}{4}

\int 30 x^{2} d x = 10 x^{3}

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

= \frac{19683 x ^{12}}{4} - \frac{196830 x ^{11}}{11} + \frac{59049 x ^{10}}{2} - 29160 x^{9} + \frac{76545 x ^{8}}{4} - 8748 x^{7} + 2835 x^{6} - 648 x^{5} + \frac{405 x ^{4}}{4} - 10 x^{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{19683 x ^{12}}{4} - \frac{196830 x ^{11}}{11} + \frac{59049 x ^{10}}{2} - 29160 x^{9} + \frac{76545 x ^{8}}{4} - 8748 x^{7} + 2835 x^{6} - 648 x^{5} + \frac{405 x ^{4}}{4} - 10 x^{3} + \frac{x ^{2}}{2} + C

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Frequently Asked Questions (FAQ)

What is the integral of x(3x-1)^{10} ?
The integral of x(3x-1)^{10} is