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Popular >

integral of x^3(x^2+1)^8

Solution

\int x^{3} (x^{2} + 1)^{8} d x

Solution

\frac{x ^{20}}{20} + \frac{4 x ^{18}}{9} + \frac{7 x ^{16}}{4} + 4 x^{14} + \frac{35 x ^{12}}{6} + \frac{28 x ^{10}}{5} + \frac{7 x ^{8}}{2} + \frac{4 x ^{6}}{3} + \frac{x ^{4}}{4} + C

Solution steps

\int x^{3} (x^{2} + 1)^{8} d x

Expand

x^{3} (x^{2} + 1)^{8} : x^{19} + 8 x^{17} + 28 x^{15} + 56 x^{13} + 70 x^{11} + 56 x^{9} + 28 x^{7} + 8 x^{5} + x^{3}

= \int x^{19} + 8 x^{17} + 28 x^{15} + 56 x^{13} + 70 x^{11} + 56 x^{9} + 28 x^{7} + 8 x^{5} + x^{3} 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^{19} d x + \int 8 x^{17} d x + \int 28 x^{15} d x + \int 56 x^{13} d x + \int 70 x^{11} d x + \int 56 x^{9} d x + \int 28 x^{7} d x + \int 8 x^{5} d x + \int x^{3} d x

\int x^{19} d x = \frac{x ^{20}}{20}

\int 8 x^{17} d x = \frac{4 x ^{18}}{9}

\int 28 x^{15} d x = \frac{7 x ^{16}}{4}

\int 56 x^{13} d x = 4 x^{14}

\int 70 x^{11} d x = \frac{35 x ^{12}}{6}

\int 56 x^{9} d x = \frac{28 x ^{10}}{5}

\int 28 x^{7} d x = \frac{7 x ^{8}}{2}

\int 8 x^{5} d x = \frac{4 x ^{6}}{3}

\int x^{3} d x = \frac{x ^{4}}{4}

= \frac{x ^{20}}{20} + \frac{4 x ^{18}}{9} + \frac{7 x ^{16}}{4} + 4 x^{14} + \frac{35 x ^{12}}{6} + \frac{28 x ^{10}}{5} + \frac{7 x ^{8}}{2} + \frac{4 x ^{6}}{3} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{x ^{20}}{20} + \frac{4 x ^{18}}{9} + \frac{7 x ^{16}}{4} + 4 x^{14} + \frac{35 x ^{12}}{6} + \frac{28 x ^{10}}{5} + \frac{7 x ^{8}}{2} + \frac{4 x ^{6}}{3} + \frac{x ^{4}}{4} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^3(x^2+1)^8 ?
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