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Popular >

integral of (1+x^2)^{10}

Solution

\int (1 + x^{2})^{10} d x

Solution

x + \frac{10 x ^{3}}{3} + 9 x^{5} + \frac{120 x ^{7}}{7} + \frac{70 x ^{9}}{3} + \frac{252 x ^{11}}{11} + \frac{210 x ^{13}}{13} + 8 x^{15} + \frac{45 x ^{17}}{17} + \frac{10 x ^{19}}{19} + \frac{x ^{21}}{21} + C

Solution steps

\int (1 + x^{2})^{10} d x

Expand

(1 + x^{2})^{10} : 1 + 10 x^{2} + 45 x^{4} + 120 x^{6} + 210 x^{8} + 252 x^{10} + 210 x^{12} + 120 x^{14} + 45 x^{16} + 10 x^{18} + x^{20}

= \int 1 + 10 x^{2} + 45 x^{4} + 120 x^{6} + 210 x^{8} + 252 x^{10} + 210 x^{12} + 120 x^{14} + 45 x^{16} + 10 x^{18} + x^{20} 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 1 d x + \int 10 x^{2} d x + \int 45 x^{4} d x + \int 120 x^{6} d x + \int 210 x^{8} d x + \int 252 x^{10} d x + \int 210 x^{12} d x + \int 120 x^{14} d x + \int 45 x^{16} d x + \int 10 x^{18} d x + \int x^{20} d x

\int 1 d x = x

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

\int 45 x^{4} d x = 9 x^{5}

\int 120 x^{6} d x = \frac{120 x ^{7}}{7}

\int 210 x^{8} d x = \frac{70 x ^{9}}{3}

\int 252 x^{10} d x = \frac{252 x ^{11}}{11}

\int 210 x^{12} d x = \frac{210 x ^{13}}{13}

\int 120 x^{14} d x = 8 x^{15}

\int 45 x^{16} d x = \frac{45 x ^{17}}{17}

\int 10 x^{18} d x = \frac{10 x ^{19}}{19}

\int x^{20} d x = \frac{x ^{21}}{21}

= x + \frac{10 x ^{3}}{3} + 9 x^{5} + \frac{120 x ^{7}}{7} + \frac{70 x ^{9}}{3} + \frac{252 x ^{11}}{11} + \frac{210 x ^{13}}{13} + 8 x^{15} + \frac{45 x ^{17}}{17} + \frac{10 x ^{19}}{19} + \frac{x ^{21}}{21}

Add a constant to the solution

= x + \frac{10 x ^{3}}{3} + 9 x^{5} + \frac{120 x ^{7}}{7} + \frac{70 x ^{9}}{3} + \frac{252 x ^{11}}{11} + \frac{210 x ^{13}}{13} + 8 x^{15} + \frac{45 x ^{17}}{17} + \frac{10 x ^{19}}{19} + \frac{x ^{21}}{21} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of (1+x^2)^{10} ?
The integral of (1+x^2)^{10} is