What is the integral of (x^2+4x)^{13}(2x+4) ?

The integral of (x^2+4x)^{13}(2x+4) is

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integral of (x^2+4x)^{13}(2x+4)

\int (x^{2} + 4 x)^{13} (2 x + 4) d x

\frac{x ^{28}}{14} + 4 x^{27} + 104 x^{26} + 1664 x^{25} + 18304 x^{24} + 146432 x^{23} + 878592 x^{22} + \frac{28114944 x ^{21}}{7} + 14057472 x^{20} + 37486592 x^{19} + 74973184 x^{18} + 109051904 x^{17} + 109051904 x^{16} + 67108864 x^{15} + \frac{134217728 x ^{14}}{7} + C

Solution steps

\int (x^{2} + 4 x)^{13} (2 x + 4) d x

Expand

(x^{2} + 4 x)^{13} (2 x + 4) : 2 x^{27} + 108 x^{26} + 2704 x^{25} + 41600 x^{24} + 439296 x^{23} + 3367936 x^{22} + 19329024 x^{21} + 84344832 x^{20} + 281149440 x^{19} + 712245248 x^{18} + 1349517312 x^{17} + 1853882368 x^{16} + 1744830464 x^{15} + 1006632960 x^{14} + 268435456 x^{13}

= \int 2 x^{27} + 108 x^{26} + 2704 x^{25} + 41600 x^{24} + 439296 x^{23} + 3367936 x^{22} + 19329024 x^{21} + 84344832 x^{20} + 281149440 x^{19} + 712245248 x^{18} + 1349517312 x^{17} + 1853882368 x^{16} + 1744830464 x^{15} + 1006632960 x^{14} + 268435456 x^{13} 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 2 x^{27} d x + \int 108 x^{26} d x + \int 2704 x^{25} d x + \int 41600 x^{24} d x + \int 439296 x^{23} d x + \int 3367936 x^{22} d x + \int 19329024 x^{21} d x + \int 84344832 x^{20} d x + \int 281149440 x^{19} d x + \int 712245248 x^{18} d x + \int 1349517312 x^{17} d x + \int 1853882368 x^{16} d x + \int 1744830464 x^{15} d x + \int 1006632960 x^{14} d x + \int 268435456 x^{13} d x

\int 2 x^{27} d x = \frac{x ^{28}}{14}

\int 108 x^{26} d x = 4 x^{27}

\int 2704 x^{25} d x = 104 x^{26}

\int 41600 x^{24} d x = 1664 x^{25}

\int 439296 x^{23} d x = 18304 x^{24}

\int 3367936 x^{22} d x = 146432 x^{23}

\int 19329024 x^{21} d x = 878592 x^{22}

\int 84344832 x^{20} d x = \frac{28114944 x ^{21}}{7}

\int 281149440 x^{19} d x = 14057472 x^{20}

\int 712245248 x^{18} d x = 37486592 x^{19}

\int 1349517312 x^{17} d x = 74973184 x^{18}

\int 1853882368 x^{16} d x = 109051904 x^{17}

\int 1744830464 x^{15} d x = 109051904 x^{16}

\int 1006632960 x^{14} d x = 67108864 x^{15}

\int 268435456 x^{13} d x = \frac{134217728 x ^{14}}{7}

= \frac{x ^{28}}{14} + 4 x^{27} + 104 x^{26} + 1664 x^{25} + 18304 x^{24} + 146432 x^{23} + 878592 x^{22} + \frac{28114944 x ^{21}}{7} + 14057472 x^{20} + 37486592 x^{19} + 74973184 x^{18} + 109051904 x^{17} + 109051904 x^{16} + 67108864 x^{15} + \frac{134217728 x ^{14}}{7}

Add a constant to the solution

= \frac{x ^{28}}{14} + 4 x^{27} + 104 x^{26} + 1664 x^{25} + 18304 x^{24} + 146432 x^{23} + 878592 x^{22} + \frac{28114944 x ^{21}}{7} + 14057472 x^{20} + 37486592 x^{19} + 74973184 x^{18} + 109051904 x^{17} + 109051904 x^{16} + 67108864 x^{15} + \frac{134217728 x ^{14}}{7} + C

\int \frac{3 x ^{2} - 1}{x ^{3} - x + 3} d x

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What is the integral of (x^2+4x)^{13}(2x+4) ?
The integral of (x^2+4x)^{13}(2x+4) is