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Popular >

integral of x^4(x^5+1)^9

Solution

\int x^{4} (x^{5} + 1)^{9} d x

Solution

\frac{x ^{50}}{50} + \frac{x ^{45}}{5} + \frac{9 x ^{40}}{10} + \frac{12 x ^{35}}{5} + \frac{21 x ^{30}}{5} + \frac{126 x ^{25}}{25} + \frac{21 x ^{20}}{5} + \frac{12 x ^{15}}{5} + \frac{9 x ^{10}}{10} + \frac{x ^{5}}{5} + C

Solution steps

\int x^{4} (x^{5} + 1)^{9} d x

Expand

x^{4} (x^{5} + 1)^{9} : x^{49} + 9 x^{44} + 36 x^{39} + 84 x^{34} + 126 x^{29} + 126 x^{24} + 84 x^{19} + 36 x^{14} + 9 x^{9} + x^{4}

= \int x^{49} + 9 x^{44} + 36 x^{39} + 84 x^{34} + 126 x^{29} + 126 x^{24} + 84 x^{19} + 36 x^{14} + 9 x^{9} + 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^{49} d x + \int 9 x^{44} d x + \int 36 x^{39} d x + \int 84 x^{34} d x + \int 126 x^{29} d x + \int 126 x^{24} d x + \int 84 x^{19} d x + \int 36 x^{14} d x + \int 9 x^{9} d x + \int x^{4} d x

\int x^{49} d x = \frac{x ^{50}}{50}

\int 9 x^{44} d x = \frac{x ^{45}}{5}

\int 36 x^{39} d x = \frac{9 x ^{40}}{10}

\int 84 x^{34} d x = \frac{12 x ^{35}}{5}

\int 126 x^{29} d x = \frac{21 x ^{30}}{5}

\int 126 x^{24} d x = \frac{126 x ^{25}}{25}

\int 84 x^{19} d x = \frac{21 x ^{20}}{5}

\int 36 x^{14} d x = \frac{12 x ^{15}}{5}

\int 9 x^{9} d x = \frac{9 x ^{10}}{10}

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

= \frac{x ^{50}}{50} + \frac{x ^{45}}{5} + \frac{9 x ^{40}}{10} + \frac{12 x ^{35}}{5} + \frac{21 x ^{30}}{5} + \frac{126 x ^{25}}{25} + \frac{21 x ^{20}}{5} + \frac{12 x ^{15}}{5} + \frac{9 x ^{10}}{10} + \frac{x ^{5}}{5}

Add a constant to the solution

= \frac{x ^{50}}{50} + \frac{x ^{45}}{5} + \frac{9 x ^{40}}{10} + \frac{12 x ^{35}}{5} + \frac{21 x ^{30}}{5} + \frac{126 x ^{25}}{25} + \frac{21 x ^{20}}{5} + \frac{12 x ^{15}}{5} + \frac{9 x ^{10}}{10} + \frac{x ^{5}}{5} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^4(x^5+1)^9 ?
The integral of x^4(x^5+1)^9 is