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Popular >

integral of x^4(x^5-4)^8

Solution

\int x^{4} (x^{5} - 4)^{8} d x

Solution

\frac{x ^{45}}{45} - \frac{4 x ^{40}}{5} + \frac{64 x ^{35}}{5} - \frac{1792 x ^{30}}{15} + \frac{3584 x ^{25}}{5} - \frac{14336 x ^{20}}{5} + \frac{114688 x ^{15}}{15} - \frac{65536 x ^{10}}{5} + \frac{65536 x ^{5}}{5} + C

Solution steps

\int x^{4} (x^{5} - 4)^{8} d x

Expand

x^{4} (x^{5} - 4)^{8} : x^{44} - 32 x^{39} + 448 x^{34} - 3584 x^{29} + 17920 x^{24} - 57344 x^{19} + 114688 x^{14} - 131072 x^{9} + 65536 x^{4}

= \int x^{44} - 32 x^{39} + 448 x^{34} - 3584 x^{29} + 17920 x^{24} - 57344 x^{19} + 114688 x^{14} - 131072 x^{9} + 65536 x^{4} 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^{44} d x - \int 32 x^{39} d x + \int 448 x^{34} d x - \int 3584 x^{29} d x + \int 17920 x^{24} d x - \int 57344 x^{19} d x + \int 114688 x^{14} d x - \int 131072 x^{9} d x + \int 65536 x^{4} d x

\int x^{44} d x = \frac{x ^{45}}{45}

\int 32 x^{39} d x = \frac{4 x ^{40}}{5}

\int 448 x^{34} d x = \frac{64 x ^{35}}{5}

\int 3584 x^{29} d x = \frac{1792 x ^{30}}{15}

\int 17920 x^{24} d x = \frac{3584 x ^{25}}{5}

\int 57344 x^{19} d x = \frac{14336 x ^{20}}{5}

\int 114688 x^{14} d x = \frac{114688 x ^{15}}{15}

\int 131072 x^{9} d x = \frac{65536 x ^{10}}{5}

\int 65536 x^{4} d x = \frac{65536 x ^{5}}{5}

= \frac{x ^{45}}{45} - \frac{4 x ^{40}}{5} + \frac{64 x ^{35}}{5} - \frac{1792 x ^{30}}{15} + \frac{3584 x ^{25}}{5} - \frac{14336 x ^{20}}{5} + \frac{114688 x ^{15}}{15} - \frac{65536 x ^{10}}{5} + \frac{65536 x ^{5}}{5}

Add a constant to the solution

= \frac{x ^{45}}{45} - \frac{4 x ^{40}}{5} + \frac{64 x ^{35}}{5} - \frac{1792 x ^{30}}{15} + \frac{3584 x ^{25}}{5} - \frac{14336 x ^{20}}{5} + \frac{114688 x ^{15}}{15} - \frac{65536 x ^{10}}{5} + \frac{65536 x ^{5}}{5} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^4(x^5-4)^8 ?
The integral of x^4(x^5-4)^8 is