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Popular >

integral of x^2(x^3+8)^{11}

Solution

\int x^{2} (x^{3} + 8)^{11} d x

Solution

\frac{x ^{36}}{36} + \frac{8 x ^{33}}{3} + \frac{352 x ^{30}}{3} + \frac{28160 x ^{27}}{9} + 56320 x^{24} + 720896 x^{21} + \frac{20185088 x ^{18}}{3} + 46137344 x^{15} + 230686720 x^{12} + \frac{7381975040 x ^{9}}{9} + \frac{5905580032 x ^{6}}{3} + \frac{8 ^{11} x ^{3}}{3} + C

Solution steps

\int x^{2} (x^{3} + 8)^{11} d x

Expand

x^{2} (x^{3} + 8)^{11} : x^{35} + 88 x^{32} + 3520 x^{29} + 84480 x^{26} + 1351680 x^{23} + 15138816 x^{20} + 121110528 x^{17} + 692060160 x^{14} + 2768240640 x^{11} + 7381975040 x^{8} + 8^{10} \cdot 11 x^{5} + 8^{11} x^{2}

= \int x^{35} + 88 x^{32} + 3520 x^{29} + 84480 x^{26} + 1351680 x^{23} + 15138816 x^{20} + 121110528 x^{17} + 692060160 x^{14} + 2768240640 x^{11} + 7381975040 x^{8} + 8^{10} \cdot 11 x^{5} + 8^{11} 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^{35} d x + \int 88 x^{32} d x + \int 3520 x^{29} d x + \int 84480 x^{26} d x + \int 1351680 x^{23} d x + \int 15138816 x^{20} d x + \int 121110528 x^{17} d x + \int 692060160 x^{14} d x + \int 2768240640 x^{11} d x + \int 7381975040 x^{8} d x + \int 8^{10} \cdot 11 x^{5} d x + \int 8^{11} x^{2} d x

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

\int 88 x^{32} d x = \frac{8 x ^{33}}{3}

\int 3520 x^{29} d x = \frac{352 x ^{30}}{3}

\int 84480 x^{26} d x = \frac{28160 x ^{27}}{9}

\int 1351680 x^{23} d x = 56320 x^{24}

\int 15138816 x^{20} d x = 720896 x^{21}

\int 121110528 x^{17} d x = \frac{20185088 x ^{18}}{3}

\int 692060160 x^{14} d x = 46137344 x^{15}

\int 2768240640 x^{11} d x = 230686720 x^{12}

\int 7381975040 x^{8} d x = \frac{7381975040 x ^{9}}{9}

\int 8^{10} \cdot 11 x^{5} d x = \frac{5905580032 x ^{6}}{3}

\int 8^{11} x^{2} d x = \frac{8 ^{11} x ^{3}}{3}

= \frac{x ^{36}}{36} + \frac{8 x ^{33}}{3} + \frac{352 x ^{30}}{3} + \frac{28160 x ^{27}}{9} + 56320 x^{24} + 720896 x^{21} + \frac{20185088 x ^{18}}{3} + 46137344 x^{15} + 230686720 x^{12} + \frac{7381975040 x ^{9}}{9} + \frac{5905580032 x ^{6}}{3} + \frac{8 ^{11} x ^{3}}{3}

Add a constant to the solution

= \frac{x ^{36}}{36} + \frac{8 x ^{33}}{3} + \frac{352 x ^{30}}{3} + \frac{28160 x ^{27}}{9} + 56320 x^{24} + 720896 x^{21} + \frac{20185088 x ^{18}}{3} + 46137344 x^{15} + 230686720 x^{12} + \frac{7381975040 x ^{9}}{9} + \frac{5905580032 x ^{6}}{3} + \frac{8 ^{11} x ^{3}}{3} + C

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Frequently Asked Questions (FAQ)

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