What is the integral of x^3(x^4+10)^{11} ?

The integral of x^3(x^4+10)^{11} is

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integral of x^3(x^4+10)^{11}

\int x^{3} (x^{4} + 10)^{11} d x

\frac{x ^{48}}{48} + \frac{5 x ^{44}}{2} + \frac{275 x ^{40}}{2} + \frac{13750 x ^{36}}{3} + 103125 x^{32} + 1650000 x^{28} + 19250000 x^{24} + 165000000 x^{20} + 1031250000 x^{16} + \frac{13750000000 x ^{12}}{3} + 13750000000 x^{8} + 25000000000 x^{4} + C

Solution steps

\int x^{3} (x^{4} + 10)^{11} d x

Expand

x^{3} (x^{4} + 10)^{11} : x^{47} + 110 x^{43} + 5500 x^{39} + 165000 x^{35} + 3300000 x^{31} + 46200000 x^{27} + 462000000 x^{23} + 3300000000 x^{19} + 16500000000 x^{15} + 55000000000 x^{11} + 1 0^{10} \cdot 11 x^{7} + 1 0^{11} x^{3}

= \int x^{47} + 110 x^{43} + 5500 x^{39} + 165000 x^{35} + 3300000 x^{31} + 46200000 x^{27} + 462000000 x^{23} + 3300000000 x^{19} + 16500000000 x^{15} + 55000000000 x^{11} + 1 0^{10} \cdot 11 x^{7} + 1 0^{11} 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^{47} d x + \int 110 x^{43} d x + \int 5500 x^{39} d x + \int 165000 x^{35} d x + \int 3300000 x^{31} d x + \int 46200000 x^{27} d x + \int 462000000 x^{23} d x + \int 3300000000 x^{19} d x + \int 16500000000 x^{15} d x + \int 55000000000 x^{11} d x + \int 1 0^{10} \cdot 11 x^{7} d x + \int 1 0^{11} x^{3} d x

\int x^{47} d x = \frac{x ^{48}}{48}

\int 110 x^{43} d x = \frac{5 x ^{44}}{2}

\int 5500 x^{39} d x = \frac{275 x ^{40}}{2}

\int 165000 x^{35} d x = \frac{13750 x ^{36}}{3}

\int 3300000 x^{31} d x = 103125 x^{32}

\int 46200000 x^{27} d x = 1650000 x^{28}

\int 462000000 x^{23} d x = 19250000 x^{24}

\int 3300000000 x^{19} d x = 165000000 x^{20}

\int 16500000000 x^{15} d x = 1031250000 x^{16}

\int 55000000000 x^{11} d x = \frac{13750000000 x ^{12}}{3}

\int 1 0^{10} \cdot 11 x^{7} d x = 13750000000 x^{8}

\int 1 0^{11} x^{3} d x = 25000000000 x^{4}

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{2} + \frac{275 x ^{40}}{2} + \frac{13750 x ^{36}}{3} + 103125 x^{32} + 1650000 x^{28} + 19250000 x^{24} + 165000000 x^{20} + 1031250000 x^{16} + \frac{13750000000 x ^{12}}{3} + 13750000000 x^{8} + 25000000000 x^{4}

Add a constant to the solution

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{2} + \frac{275 x ^{40}}{2} + \frac{13750 x ^{36}}{3} + 103125 x^{32} + 1650000 x^{28} + 19250000 x^{24} + 165000000 x^{20} + 1031250000 x^{16} + \frac{13750000000 x ^{12}}{3} + 13750000000 x^{8} + 25000000000 x^{4} + C