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Popular >

integral of x^6(x^7-8)^{10}

Solution

\int x^{6} (x^{7} - 8)^{10} d x

Solution

\frac{x ^{77}}{77} - \frac{8 x ^{70}}{7} + \frac{320 x ^{63}}{7} - \frac{7680 x ^{56}}{7} + \frac{122880 x ^{49}}{7} - 196608 x^{42} + 1572864 x^{35} - \frac{62914560 x ^{28}}{7} + \frac{251658240 x ^{21}}{7} - \frac{671088640 x ^{14}}{7} + \frac{8 ^{10} x ^{7}}{7} + C

Solution steps

\int x^{6} (x^{7} - 8)^{10} d x

Expand

x^{6} (x^{7} - 8)^{10} : x^{76} - 80 x^{69} + 2880 x^{62} - 61440 x^{55} + 860160 x^{48} - 8257536 x^{41} + 55050240 x^{34} - 251658240 x^{27} + 754974720 x^{20} - 1342177280 x^{13} + 8^{10} x^{6}

= \int x^{76} - 80 x^{69} + 2880 x^{62} - 61440 x^{55} + 860160 x^{48} - 8257536 x^{41} + 55050240 x^{34} - 251658240 x^{27} + 754974720 x^{20} - 1342177280 x^{13} + 8^{10} x^{6} 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^{76} d x - \int 80 x^{69} d x + \int 2880 x^{62} d x - \int 61440 x^{55} d x + \int 860160 x^{48} d x - \int 8257536 x^{41} d x + \int 55050240 x^{34} d x - \int 251658240 x^{27} d x + \int 754974720 x^{20} d x - \int 1342177280 x^{13} d x + \int 8^{10} x^{6} d x

\int x^{76} d x = \frac{x ^{77}}{77}

\int 80 x^{69} d x = \frac{8 x ^{70}}{7}

\int 2880 x^{62} d x = \frac{320 x ^{63}}{7}

\int 61440 x^{55} d x = \frac{7680 x ^{56}}{7}

\int 860160 x^{48} d x = \frac{122880 x ^{49}}{7}

\int 8257536 x^{41} d x = 196608 x^{42}

\int 55050240 x^{34} d x = 1572864 x^{35}

\int 251658240 x^{27} d x = \frac{62914560 x ^{28}}{7}

\int 754974720 x^{20} d x = \frac{251658240 x ^{21}}{7}

\int 1342177280 x^{13} d x = \frac{671088640 x ^{14}}{7}

\int 8^{10} x^{6} d x = \frac{8 ^{10} x ^{7}}{7}

= \frac{x ^{77}}{77} - \frac{8 x ^{70}}{7} + \frac{320 x ^{63}}{7} - \frac{7680 x ^{56}}{7} + \frac{122880 x ^{49}}{7} - 196608 x^{42} + 1572864 x^{35} - \frac{62914560 x ^{28}}{7} + \frac{251658240 x ^{21}}{7} - \frac{671088640 x ^{14}}{7} + \frac{8 ^{10} x ^{7}}{7}

Add a constant to the solution

= \frac{x ^{77}}{77} - \frac{8 x ^{70}}{7} + \frac{320 x ^{63}}{7} - \frac{7680 x ^{56}}{7} + \frac{122880 x ^{49}}{7} - 196608 x^{42} + 1572864 x^{35} - \frac{62914560 x ^{28}}{7} + \frac{251658240 x ^{21}}{7} - \frac{671088640 x ^{14}}{7} + \frac{8 ^{10} x ^{7}}{7} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^6(x^7-8)^{10} ?
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