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Popular >

integral of sec^{22}(x)

Solution

integral

sec^{22} (x)

Solution

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

Solution steps

\int sec^{22} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

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

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

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

\int 1 d u = u

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

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

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

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

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

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

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

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

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

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

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

Substitute back

u = tan (x)

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

Add a constant to the solution

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

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Frequently Asked Questions (FAQ)

What is the integral of sec^{22}(x) ?
The integral of sec^{22}(x) is