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Popular >

integral of x(x+3)^{10}

Solution

\int x (x + 3)^{10} d x

Solution

\frac{x ^{12}}{12} + \frac{30 x ^{11}}{11} + \frac{81 x ^{10}}{2} + 360 x^{9} + \frac{8505 x ^{8}}{4} + 8748 x^{7} + 25515 x^{6} + 52488 x^{5} + \frac{295245 x ^{4}}{4} + 65610 x^{3} + \frac{59049 x ^{2}}{2} + C

Solution steps

\int x (x + 3)^{10} d x

Expand

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

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

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

\int 30 x^{10} d x = \frac{30 x ^{11}}{11}

\int 405 x^{9} d x = \frac{81 x ^{10}}{2}

\int 3240 x^{8} d x = 360 x^{9}

\int 17010 x^{7} d x = \frac{8505 x ^{8}}{4}

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

\int 153090 x^{5} d x = 25515 x^{6}

\int 262440 x^{4} d x = 52488 x^{5}

\int 295245 x^{3} d x = \frac{295245 x ^{4}}{4}

\int 196830 x^{2} d x = 65610 x^{3}

\int 59049 x d x = \frac{59049 x ^{2}}{2}

= \frac{x ^{12}}{12} + \frac{30 x ^{11}}{11} + \frac{81 x ^{10}}{2} + 360 x^{9} + \frac{8505 x ^{8}}{4} + 8748 x^{7} + 25515 x^{6} + 52488 x^{5} + \frac{295245 x ^{4}}{4} + 65610 x^{3} + \frac{59049 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{12}}{12} + \frac{30 x ^{11}}{11} + \frac{81 x ^{10}}{2} + 360 x^{9} + \frac{8505 x ^{8}}{4} + 8748 x^{7} + 25515 x^{6} + 52488 x^{5} + \frac{295245 x ^{4}}{4} + 65610 x^{3} + \frac{59049 x ^{2}}{2} + C

Graph

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Frequently Asked Questions (FAQ)

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