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Popular >

integral of x^3(x^2+1)^9

Solution

\int x^{3} (x^{2} + 1)^{9} d x

Solution

\frac{x ^{22}}{22} + \frac{9 x ^{20}}{20} + 2 x^{18} + \frac{21 x ^{16}}{4} + 9 x^{14} + \frac{21 x ^{12}}{2} + \frac{42 x ^{10}}{5} + \frac{9 x ^{8}}{2} + \frac{3 x ^{6}}{2} + \frac{x ^{4}}{4} + C

Solution steps

\int x^{3} (x^{2} + 1)^{9} d x

Expand

x^{3} (x^{2} + 1)^{9} : x^{21} + 9 x^{19} + 36 x^{17} + 84 x^{15} + 126 x^{13} + 126 x^{11} + 84 x^{9} + 36 x^{7} + 9 x^{5} + x^{3}

= \int x^{21} + 9 x^{19} + 36 x^{17} + 84 x^{15} + 126 x^{13} + 126 x^{11} + 84 x^{9} + 36 x^{7} + 9 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^{21} d x + \int 9 x^{19} d x + \int 36 x^{17} d x + \int 84 x^{15} d x + \int 126 x^{13} d x + \int 126 x^{11} d x + \int 84 x^{9} d x + \int 36 x^{7} d x + \int 9 x^{5} d x + \int x^{3} d x

\int x^{21} d x = \frac{x ^{22}}{22}

\int 9 x^{19} d x = \frac{9 x ^{20}}{20}

\int 36 x^{17} d x = 2 x^{18}

\int 84 x^{15} d x = \frac{21 x ^{16}}{4}

\int 126 x^{13} d x = 9 x^{14}

\int 126 x^{11} d x = \frac{21 x ^{12}}{2}

\int 84 x^{9} d x = \frac{42 x ^{10}}{5}

\int 36 x^{7} d x = \frac{9 x ^{8}}{2}

\int 9 x^{5} d x = \frac{3 x ^{6}}{2}

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

= \frac{x ^{22}}{22} + \frac{9 x ^{20}}{20} + 2 x^{18} + \frac{21 x ^{16}}{4} + 9 x^{14} + \frac{21 x ^{12}}{2} + \frac{42 x ^{10}}{5} + \frac{9 x ^{8}}{2} + \frac{3 x ^{6}}{2} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{x ^{22}}{22} + \frac{9 x ^{20}}{20} + 2 x^{18} + \frac{21 x ^{16}}{4} + 9 x^{14} + \frac{21 x ^{12}}{2} + \frac{42 x ^{10}}{5} + \frac{9 x ^{8}}{2} + \frac{3 x ^{6}}{2} + \frac{x ^{4}}{4} + C

Popular Examples

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\int \frac{1}{2 x - 3} d x

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\int \frac{x ^{2}}{( 4 x ^{2} + 9 ) ^{2}} d x

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\int \frac{5}{6 + 7 x ^{2}} d x

integral of 2/x+sqrt(x)

\int \frac{2}{x} + x d x

integral of e^{3x^2}2x

\int e^{3 x^{2}} 2 x d x

Frequently Asked Questions (FAQ)

What is the integral of x^3(x^2+1)^9 ?
The integral of x^3(x^2+1)^9 is