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Popular >

integral of x^2(x^3+8)^{10}

Solution

\int x^{2} (x^{3} + 8)^{10} d x

Solution

\frac{x ^{33}}{33} + \frac{8 x ^{30}}{3} + \frac{320 x ^{27}}{3} + 2560 x^{24} + 40960 x^{21} + 458752 x^{18} + 3670016 x^{15} + 20971520 x^{12} + 83886080 x^{9} + \frac{671088640 x ^{6}}{3} + \frac{8 ^{10} x ^{3}}{3} + C

Solution steps

\int x^{2} (x^{3} + 8)^{10} d x

Expand

x^{2} (x^{3} + 8)^{10} : x^{32} + 80 x^{29} + 2880 x^{26} + 61440 x^{23} + 860160 x^{20} + 8257536 x^{17} + 55050240 x^{14} + 251658240 x^{11} + 754974720 x^{8} + 1342177280 x^{5} + 8^{10} x^{2}

= \int x^{32} + 80 x^{29} + 2880 x^{26} + 61440 x^{23} + 860160 x^{20} + 8257536 x^{17} + 55050240 x^{14} + 251658240 x^{11} + 754974720 x^{8} + 1342177280 x^{5} + 8^{10} x^{2} 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^{32} d x + \int 80 x^{29} d x + \int 2880 x^{26} d x + \int 61440 x^{23} d x + \int 860160 x^{20} d x + \int 8257536 x^{17} d x + \int 55050240 x^{14} d x + \int 251658240 x^{11} d x + \int 754974720 x^{8} d x + \int 1342177280 x^{5} d x + \int 8^{10} x^{2} d x

\int x^{32} d x = \frac{x ^{33}}{33}

\int 80 x^{29} d x = \frac{8 x ^{30}}{3}

\int 2880 x^{26} d x = \frac{320 x ^{27}}{3}

\int 61440 x^{23} d x = 2560 x^{24}

\int 860160 x^{20} d x = 40960 x^{21}

\int 8257536 x^{17} d x = 458752 x^{18}

\int 55050240 x^{14} d x = 3670016 x^{15}

\int 251658240 x^{11} d x = 20971520 x^{12}

\int 754974720 x^{8} d x = 83886080 x^{9}

\int 1342177280 x^{5} d x = \frac{671088640 x ^{6}}{3}

\int 8^{10} x^{2} d x = \frac{8 ^{10} x ^{3}}{3}

= \frac{x ^{33}}{33} + \frac{8 x ^{30}}{3} + \frac{320 x ^{27}}{3} + 2560 x^{24} + 40960 x^{21} + 458752 x^{18} + 3670016 x^{15} + 20971520 x^{12} + 83886080 x^{9} + \frac{671088640 x ^{6}}{3} + \frac{8 ^{10} x ^{3}}{3}

Add a constant to the solution

= \frac{x ^{33}}{33} + \frac{8 x ^{30}}{3} + \frac{320 x ^{27}}{3} + 2560 x^{24} + 40960 x^{21} + 458752 x^{18} + 3670016 x^{15} + 20971520 x^{12} + 83886080 x^{9} + \frac{671088640 x ^{6}}{3} + \frac{8 ^{10} x ^{3}}{3} + C

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Frequently Asked Questions (FAQ)

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