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Popular Calculus >

integral of x^3(x^2+1)^{13}

Solution

\int x^{3} (x^{2} + 1)^{13} d x

Solution

\frac{x ^{30}}{30} + \frac{13 x ^{28}}{28} + 3 x^{26} + \frac{143 x ^{24}}{12} + \frac{65 x ^{22}}{2} + \frac{1287 x ^{20}}{20} + \frac{286 x ^{18}}{3} + \frac{429 x ^{16}}{4} + \frac{1287 x ^{14}}{14} + \frac{715 x ^{12}}{12} + \frac{143 x ^{10}}{5} + \frac{39 x ^{8}}{4} + \frac{13 x ^{6}}{6} + \frac{x ^{4}}{4} + C

Solution steps

\int x^{3} (x^{2} + 1)^{13} d x

Expand

x^{3} (x^{2} + 1)^{13} : x^{29} + 13 x^{27} + 78 x^{25} + 286 x^{23} + 715 x^{21} + 1287 x^{19} + 1716 x^{17} + 1716 x^{15} + 1287 x^{13} + 715 x^{11} + 286 x^{9} + 78 x^{7} + 13 x^{5} + x^{3}

= \int x^{29} + 13 x^{27} + 78 x^{25} + 286 x^{23} + 715 x^{21} + 1287 x^{19} + 1716 x^{17} + 1716 x^{15} + 1287 x^{13} + 715 x^{11} + 286 x^{9} + 78 x^{7} + 13 x^{5} + x^{3} 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^{29} d x + \int 13 x^{27} d x + \int 78 x^{25} d x + \int 286 x^{23} d x + \int 715 x^{21} d x + \int 1287 x^{19} d x + \int 1716 x^{17} d x + \int 1716 x^{15} d x + \int 1287 x^{13} d x + \int 715 x^{11} d x + \int 286 x^{9} d x + \int 78 x^{7} d x + \int 13 x^{5} d x + \int x^{3} d x

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

\int 13 x^{27} d x = \frac{13 x ^{28}}{28}

\int 78 x^{25} d x = 3 x^{26}

\int 286 x^{23} d x = \frac{143 x ^{24}}{12}

\int 715 x^{21} d x = \frac{65 x ^{22}}{2}

\int 1287 x^{19} d x = \frac{1287 x ^{20}}{20}

\int 1716 x^{17} d x = \frac{286 x ^{18}}{3}

\int 1716 x^{15} d x = \frac{429 x ^{16}}{4}

\int 1287 x^{13} d x = \frac{1287 x ^{14}}{14}

\int 715 x^{11} d x = \frac{715 x ^{12}}{12}

\int 286 x^{9} d x = \frac{143 x ^{10}}{5}

\int 78 x^{7} d x = \frac{39 x ^{8}}{4}

\int 13 x^{5} d x = \frac{13 x ^{6}}{6}

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

= \frac{x ^{30}}{30} + \frac{13 x ^{28}}{28} + 3 x^{26} + \frac{143 x ^{24}}{12} + \frac{65 x ^{22}}{2} + \frac{1287 x ^{20}}{20} + \frac{286 x ^{18}}{3} + \frac{429 x ^{16}}{4} + \frac{1287 x ^{14}}{14} + \frac{715 x ^{12}}{12} + \frac{143 x ^{10}}{5} + \frac{39 x ^{8}}{4} + \frac{13 x ^{6}}{6} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{x ^{30}}{30} + \frac{13 x ^{28}}{28} + 3 x^{26} + \frac{143 x ^{24}}{12} + \frac{65 x ^{22}}{2} + \frac{1287 x ^{20}}{20} + \frac{286 x ^{18}}{3} + \frac{429 x ^{16}}{4} + \frac{1287 x ^{14}}{14} + \frac{715 x ^{12}}{12} + \frac{143 x ^{10}}{5} + \frac{39 x ^{8}}{4} + \frac{13 x ^{6}}{6} + \frac{x ^{4}}{4} + C

Graph

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Frequently Asked Questions (FAQ)

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