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Popular >

integral of x^3(5+x^4)^{11}

Solution

\int x^{3} (5 + x^{4})^{11} d x

Solution

\frac{48828125 x ^{4}}{4} + \frac{107421875 x ^{8}}{8} + \frac{107421875 x ^{12}}{12} + \frac{64453125 x ^{16}}{16} + \frac{2578125 x ^{20}}{2} + \frac{1203125 x ^{24}}{4} + \frac{103125 x ^{28}}{2} + \frac{103125 x ^{32}}{16} + \frac{6875 x ^{36}}{12} + \frac{275 x ^{40}}{8} + \frac{5 x ^{44}}{4} + \frac{x ^{48}}{48} + C

Solution steps

\int x^{3} (5 + x^{4})^{11} d x

Expand

x^{3} (5 + x^{4})^{11} : 48828125 x^{3} + 107421875 x^{7} + 107421875 x^{11} + 64453125 x^{15} + 25781250 x^{19} + 7218750 x^{23} + 1443750 x^{27} + 206250 x^{31} + 20625 x^{35} + 1375 x^{39} + 55 x^{43} + x^{47}

= \int 48828125 x^{3} + 107421875 x^{7} + 107421875 x^{11} + 64453125 x^{15} + 25781250 x^{19} + 7218750 x^{23} + 1443750 x^{27} + 206250 x^{31} + 20625 x^{35} + 1375 x^{39} + 55 x^{43} + x^{47} 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 48828125 x^{3} d x + \int 107421875 x^{7} d x + \int 107421875 x^{11} d x + \int 64453125 x^{15} d x + \int 25781250 x^{19} d x + \int 7218750 x^{23} d x + \int 1443750 x^{27} d x + \int 206250 x^{31} d x + \int 20625 x^{35} d x + \int 1375 x^{39} d x + \int 55 x^{43} d x + \int x^{47} d x

\int 48828125 x^{3} d x = \frac{48828125 x ^{4}}{4}

\int 107421875 x^{7} d x = \frac{107421875 x ^{8}}{8}

\int 107421875 x^{11} d x = \frac{107421875 x ^{12}}{12}

\int 64453125 x^{15} d x = \frac{64453125 x ^{16}}{16}

\int 25781250 x^{19} d x = \frac{2578125 x ^{20}}{2}

\int 7218750 x^{23} d x = \frac{1203125 x ^{24}}{4}

\int 1443750 x^{27} d x = \frac{103125 x ^{28}}{2}

\int 206250 x^{31} d x = \frac{103125 x ^{32}}{16}

\int 20625 x^{35} d x = \frac{6875 x ^{36}}{12}

\int 1375 x^{39} d x = \frac{275 x ^{40}}{8}

\int 55 x^{43} d x = \frac{5 x ^{44}}{4}

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

= \frac{48828125 x ^{4}}{4} + \frac{107421875 x ^{8}}{8} + \frac{107421875 x ^{12}}{12} + \frac{64453125 x ^{16}}{16} + \frac{2578125 x ^{20}}{2} + \frac{1203125 x ^{24}}{4} + \frac{103125 x ^{28}}{2} + \frac{103125 x ^{32}}{16} + \frac{6875 x ^{36}}{12} + \frac{275 x ^{40}}{8} + \frac{5 x ^{44}}{4} + \frac{x ^{48}}{48}

Add a constant to the solution

= \frac{48828125 x ^{4}}{4} + \frac{107421875 x ^{8}}{8} + \frac{107421875 x ^{12}}{12} + \frac{64453125 x ^{16}}{16} + \frac{2578125 x ^{20}}{2} + \frac{1203125 x ^{24}}{4} + \frac{103125 x ^{28}}{2} + \frac{103125 x ^{32}}{16} + \frac{6875 x ^{36}}{12} + \frac{275 x ^{40}}{8} + \frac{5 x ^{44}}{4} + \frac{x ^{48}}{48} + C

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Frequently Asked Questions (FAQ)

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