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Popular >

integral of x^8(x^9-10)^7

Solution

\int x^{8} (x^{9} - 10)^{7} d x

Solution

\frac{x ^{72}}{72} - \frac{10 x ^{63}}{9} + \frac{350 x ^{54}}{9} - \frac{7000 x ^{45}}{9} + \frac{87500 x ^{36}}{9} - \frac{700000 x ^{27}}{9} + \frac{3500000 x ^{18}}{9} - \frac{10000000 x ^{9}}{9} + C

Solution steps

\int x^{8} (x^{9} - 10)^{7} d x

Expand

x^{8} (x^{9} - 10)^{7} : x^{71} - 70 x^{62} + 2100 x^{53} - 35000 x^{44} + 350000 x^{35} - 2100000 x^{26} + 7000000 x^{17} - 10000000 x^{8}

= \int x^{71} - 70 x^{62} + 2100 x^{53} - 35000 x^{44} + 350000 x^{35} - 2100000 x^{26} + 7000000 x^{17} - 10000000 x^{8} 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^{71} d x - \int 70 x^{62} d x + \int 2100 x^{53} d x - \int 35000 x^{44} d x + \int 350000 x^{35} d x - \int 2100000 x^{26} d x + \int 7000000 x^{17} d x - \int 10000000 x^{8} d x

\int x^{71} d x = \frac{x ^{72}}{72}

\int 70 x^{62} d x = \frac{10 x ^{63}}{9}

\int 2100 x^{53} d x = \frac{350 x ^{54}}{9}

\int 35000 x^{44} d x = \frac{7000 x ^{45}}{9}

\int 350000 x^{35} d x = \frac{87500 x ^{36}}{9}

\int 2100000 x^{26} d x = \frac{700000 x ^{27}}{9}

\int 7000000 x^{17} d x = \frac{3500000 x ^{18}}{9}

\int 10000000 x^{8} d x = \frac{10000000 x ^{9}}{9}

= \frac{x ^{72}}{72} - \frac{10 x ^{63}}{9} + \frac{350 x ^{54}}{9} - \frac{7000 x ^{45}}{9} + \frac{87500 x ^{36}}{9} - \frac{700000 x ^{27}}{9} + \frac{3500000 x ^{18}}{9} - \frac{10000000 x ^{9}}{9}

Add a constant to the solution

= \frac{x ^{72}}{72} - \frac{10 x ^{63}}{9} + \frac{350 x ^{54}}{9} - \frac{7000 x ^{45}}{9} + \frac{87500 x ^{36}}{9} - \frac{700000 x ^{27}}{9} + \frac{3500000 x ^{18}}{9} - \frac{10000000 x ^{9}}{9} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^8(x^9-10)^7 ?
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