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Popular >

integral of x(2x+5)^{12}

Solution

\int x (2 x + 5)^{12} d x

Solution

\frac{2048 x ^{14}}{7} + \frac{122880 x ^{13}}{13} + 140800 x^{12} + 1280000 x^{11} + 7920000 x^{10} + 35200000 x^{9} + 115500000 x^{8} + \frac{1980000000 x ^{7}}{7} + 515625000 x^{6} + 687500000 x^{5} + 644531250 x^{4} + 390625000 x^{3} + \frac{244140625 x ^{2}}{2} + C

Solution steps

\int x (2 x + 5)^{12} d x

Expand

x (2 x + 5)^{12} : 4096 x^{13} + 122880 x^{12} + 1689600 x^{11} + 14080000 x^{10} + 79200000 x^{9} + 316800000 x^{8} + 924000000 x^{7} + 1980000000 x^{6} + 3093750000 x^{5} + 3437500000 x^{4} + 2578125000 x^{3} + 1171875000 x^{2} + 244140625 x

= \int 4096 x^{13} + 122880 x^{12} + 1689600 x^{11} + 14080000 x^{10} + 79200000 x^{9} + 316800000 x^{8} + 924000000 x^{7} + 1980000000 x^{6} + 3093750000 x^{5} + 3437500000 x^{4} + 2578125000 x^{3} + 1171875000 x^{2} + 244140625 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 4096 x^{13} d x + \int 122880 x^{12} d x + \int 1689600 x^{11} d x + \int 14080000 x^{10} d x + \int 79200000 x^{9} d x + \int 316800000 x^{8} d x + \int 924000000 x^{7} d x + \int 1980000000 x^{6} d x + \int 3093750000 x^{5} d x + \int 3437500000 x^{4} d x + \int 2578125000 x^{3} d x + \int 1171875000 x^{2} d x + \int 244140625 x d x

\int 4096 x^{13} d x = \frac{2048 x ^{14}}{7}

\int 122880 x^{12} d x = \frac{122880 x ^{13}}{13}

\int 1689600 x^{11} d x = 140800 x^{12}

\int 14080000 x^{10} d x = 1280000 x^{11}

\int 79200000 x^{9} d x = 7920000 x^{10}

\int 316800000 x^{8} d x = 35200000 x^{9}

\int 924000000 x^{7} d x = 115500000 x^{8}

\int 1980000000 x^{6} d x = \frac{1980000000 x ^{7}}{7}

\int 3093750000 x^{5} d x = 515625000 x^{6}

\int 3437500000 x^{4} d x = 687500000 x^{5}

\int 2578125000 x^{3} d x = 644531250 x^{4}

\int 1171875000 x^{2} d x = 390625000 x^{3}

\int 244140625 x d x = \frac{244140625 x ^{2}}{2}

= \frac{2048 x ^{14}}{7} + \frac{122880 x ^{13}}{13} + 140800 x^{12} + 1280000 x^{11} + 7920000 x^{10} + 35200000 x^{9} + 115500000 x^{8} + \frac{1980000000 x ^{7}}{7} + 515625000 x^{6} + 687500000 x^{5} + 644531250 x^{4} + 390625000 x^{3} + \frac{244140625 x ^{2}}{2}

Add a constant to the solution

= \frac{2048 x ^{14}}{7} + \frac{122880 x ^{13}}{13} + 140800 x^{12} + 1280000 x^{11} + 7920000 x^{10} + 35200000 x^{9} + 115500000 x^{8} + \frac{1980000000 x ^{7}}{7} + 515625000 x^{6} + 687500000 x^{5} + 644531250 x^{4} + 390625000 x^{3} + \frac{244140625 x ^{2}}{2} + C

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Frequently Asked Questions (FAQ)

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