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Popular Calculus >

integral of x(x^2-2)^{15}

Solution

\int x (x^{2} - 2)^{15} d x

Solution

\frac{x ^{32}}{32} - x^{30} + 15 x^{28} - 140 x^{26} + 910 x^{24} - 4368 x^{22} + 16016 x^{20} - 45760 x^{18} + 102960 x^{16} - 183040 x^{14} + 256256 x^{12} - 279552 x^{10} + 232960 x^{8} - 143360 x^{6} + 61440 x^{4} - 16384 x^{2} + C

Solution steps

\int x (x^{2} - 2)^{15} d x

Expand

x (x^{2} - 2)^{15} : x^{31} - 30 x^{29} + 420 x^{27} - 3640 x^{25} + 21840 x^{23} - 96096 x^{21} + 320320 x^{19} - 823680 x^{17} + 1647360 x^{15} - 2562560 x^{13} + 3075072 x^{11} - 2795520 x^{9} + 1863680 x^{7} - 860160 x^{5} + 245760 x^{3} - 32768 x

= \int x^{31} - 30 x^{29} + 420 x^{27} - 3640 x^{25} + 21840 x^{23} - 96096 x^{21} + 320320 x^{19} - 823680 x^{17} + 1647360 x^{15} - 2562560 x^{13} + 3075072 x^{11} - 2795520 x^{9} + 1863680 x^{7} - 860160 x^{5} + 245760 x^{3} - 32768 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^{31} d x - \int 30 x^{29} d x + \int 420 x^{27} d x - \int 3640 x^{25} d x + \int 21840 x^{23} d x - \int 96096 x^{21} d x + \int 320320 x^{19} d x - \int 823680 x^{17} d x + \int 1647360 x^{15} d x - \int 2562560 x^{13} d x + \int 3075072 x^{11} d x - \int 2795520 x^{9} d x + \int 1863680 x^{7} d x - \int 860160 x^{5} d x + \int 245760 x^{3} d x - \int 32768 x d x

\int x^{31} d x = \frac{x ^{32}}{32}

\int 30 x^{29} d x = x^{30}

\int 420 x^{27} d x = 15 x^{28}

\int 3640 x^{25} d x = 140 x^{26}

\int 21840 x^{23} d x = 910 x^{24}

\int 96096 x^{21} d x = 4368 x^{22}

\int 320320 x^{19} d x = 16016 x^{20}

\int 823680 x^{17} d x = 45760 x^{18}

\int 1647360 x^{15} d x = 102960 x^{16}

\int 2562560 x^{13} d x = 183040 x^{14}

\int 3075072 x^{11} d x = 256256 x^{12}

\int 2795520 x^{9} d x = 279552 x^{10}

\int 1863680 x^{7} d x = 232960 x^{8}

\int 860160 x^{5} d x = 143360 x^{6}

\int 245760 x^{3} d x = 61440 x^{4}

\int 32768 x d x = 16384 x^{2}

= \frac{x ^{32}}{32} - x^{30} + 15 x^{28} - 140 x^{26} + 910 x^{24} - 4368 x^{22} + 16016 x^{20} - 45760 x^{18} + 102960 x^{16} - 183040 x^{14} + 256256 x^{12} - 279552 x^{10} + 232960 x^{8} - 143360 x^{6} + 61440 x^{4} - 16384 x^{2}

Add a constant to the solution

= \frac{x ^{32}}{32} - x^{30} + 15 x^{28} - 140 x^{26} + 910 x^{24} - 4368 x^{22} + 16016 x^{20} - 45760 x^{18} + 102960 x^{16} - 183040 x^{14} + 256256 x^{12} - 279552 x^{10} + 232960 x^{8} - 143360 x^{6} + 61440 x^{4} - 16384 x^{2} + C

Graph

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Frequently Asked Questions (FAQ)

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