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Popular Calculus >

integral of x(5x-3)^9

Solution

\int x (5 x - 3)^{9} d x

Solution

\frac{1953125 x ^{11}}{11} - \frac{2109375 x ^{10}}{2} + 2812500 x^{9} - \frac{8859375 x ^{8}}{2} + 4556250 x^{7} - 3189375 x^{6} + 1530900 x^{5} - 492075 x^{4} + 98415 x^{3} - \frac{19683 x ^{2}}{2} + C

Solution steps

\int x (5 x - 3)^{9} d x

Expand

x (5 x - 3)^{9} : 1953125 x^{10} - 10546875 x^{9} + 25312500 x^{8} - 35437500 x^{7} + 31893750 x^{6} - 19136250 x^{5} + 7654500 x^{4} - 1968300 x^{3} + 295245 x^{2} - 19683 x

= \int 1953125 x^{10} - 10546875 x^{9} + 25312500 x^{8} - 35437500 x^{7} + 31893750 x^{6} - 19136250 x^{5} + 7654500 x^{4} - 1968300 x^{3} + 295245 x^{2} - 19683 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 1953125 x^{10} d x - \int 10546875 x^{9} d x + \int 25312500 x^{8} d x - \int 35437500 x^{7} d x + \int 31893750 x^{6} d x - \int 19136250 x^{5} d x + \int 7654500 x^{4} d x - \int 1968300 x^{3} d x + \int 295245 x^{2} d x - \int 19683 x d x

\int 1953125 x^{10} d x = \frac{1953125 x ^{11}}{11}

\int 10546875 x^{9} d x = \frac{2109375 x ^{10}}{2}

\int 25312500 x^{8} d x = 2812500 x^{9}

\int 35437500 x^{7} d x = \frac{8859375 x ^{8}}{2}

\int 31893750 x^{6} d x = 4556250 x^{7}

\int 19136250 x^{5} d x = 3189375 x^{6}

\int 7654500 x^{4} d x = 1530900 x^{5}

\int 1968300 x^{3} d x = 492075 x^{4}

\int 295245 x^{2} d x = 98415 x^{3}

\int 19683 x d x = \frac{19683 x ^{2}}{2}

= \frac{1953125 x ^{11}}{11} - \frac{2109375 x ^{10}}{2} + 2812500 x^{9} - \frac{8859375 x ^{8}}{2} + 4556250 x^{7} - 3189375 x^{6} + 1530900 x^{5} - 492075 x^{4} + 98415 x^{3} - \frac{19683 x ^{2}}{2}

Add a constant to the solution

= \frac{1953125 x ^{11}}{11} - \frac{2109375 x ^{10}}{2} + 2812500 x^{9} - \frac{8859375 x ^{8}}{2} + 4556250 x^{7} - 3189375 x^{6} + 1530900 x^{5} - 492075 x^{4} + 98415 x^{3} - \frac{19683 x ^{2}}{2} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x(5x-3)^9 ?
The integral of x(5x-3)^9 is