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Popular >

integral of x^5(x^3-2)^{10}

Solution

\int x^{5} (x^{3} - 2)^{10} d x

Solution

\frac{x ^{36}}{36} - \frac{20 x ^{33}}{33} + 6 x^{30} - \frac{320 x ^{27}}{9} + 140 x^{24} - 384 x^{21} + \frac{2240 x ^{18}}{3} - 1024 x^{15} + 960 x^{12} - \frac{5120 x ^{9}}{9} + \frac{512 x ^{6}}{3} + C

Solution steps

\int x^{5} (x^{3} - 2)^{10} d x

Expand

x^{5} (x^{3} - 2)^{10} : x^{35} - 20 x^{32} + 180 x^{29} - 960 x^{26} + 3360 x^{23} - 8064 x^{20} + 13440 x^{17} - 15360 x^{14} + 11520 x^{11} - 5120 x^{8} + 1024 x^{5}

= \int x^{35} - 20 x^{32} + 180 x^{29} - 960 x^{26} + 3360 x^{23} - 8064 x^{20} + 13440 x^{17} - 15360 x^{14} + 11520 x^{11} - 5120 x^{8} + 1024 x^{5} 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^{35} d x - \int 20 x^{32} d x + \int 180 x^{29} d x - \int 960 x^{26} d x + \int 3360 x^{23} d x - \int 8064 x^{20} d x + \int 13440 x^{17} d x - \int 15360 x^{14} d x + \int 11520 x^{11} d x - \int 5120 x^{8} d x + \int 1024 x^{5} d x

\int x^{35} d x = \frac{x ^{36}}{36}

\int 20 x^{32} d x = \frac{20 x ^{33}}{33}

\int 180 x^{29} d x = 6 x^{30}

\int 960 x^{26} d x = \frac{320 x ^{27}}{9}

\int 3360 x^{23} d x = 140 x^{24}

\int 8064 x^{20} d x = 384 x^{21}

\int 13440 x^{17} d x = \frac{2240 x ^{18}}{3}

\int 15360 x^{14} d x = 1024 x^{15}

\int 11520 x^{11} d x = 960 x^{12}

\int 5120 x^{8} d x = \frac{5120 x ^{9}}{9}

\int 1024 x^{5} d x = \frac{512 x ^{6}}{3}

= \frac{x ^{36}}{36} - \frac{20 x ^{33}}{33} + 6 x^{30} - \frac{320 x ^{27}}{9} + 140 x^{24} - 384 x^{21} + \frac{2240 x ^{18}}{3} - 1024 x^{15} + 960 x^{12} - \frac{5120 x ^{9}}{9} + \frac{512 x ^{6}}{3}

Add a constant to the solution

= \frac{x ^{36}}{36} - \frac{20 x ^{33}}{33} + 6 x^{30} - \frac{320 x ^{27}}{9} + 140 x^{24} - 384 x^{21} + \frac{2240 x ^{18}}{3} - 1024 x^{15} + 960 x^{12} - \frac{5120 x ^{9}}{9} + \frac{512 x ^{6}}{3} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^5(x^3-2)^{10} ?
The integral of x^5(x^3-2)^{10} is