What is the integral of tan^{21}(x)sec^{22}(x) ?

The integral of tan^{21}(x)sec^{22}(x) is

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integral of tan^{21}(x)sec^{22}(x)

\int tan^{21} (x) sec^{22} (x) d x

\frac{tan ^{22} ( x )}{22} + \frac{5 tan ^{24} ( x )}{12} + \frac{45 tan ^{26} ( x )}{26} + \frac{30 tan ^{28} ( x )}{7} + 7 tan^{30} (x) + \frac{63 tan ^{32} ( x )}{8} + \frac{105 tan ^{34} ( x )}{17} + \frac{10 tan ^{36} ( x )}{3} + \frac{45 tan ^{38} ( x )}{38} + \frac{tan ^{40} ( x )}{4} + \frac{tan ^{42} ( x )}{42} + C

Solution steps

\int tan^{21} (x) sec^{22} (x) d x

Rewrite using trig identities

= \int sec^{2} (x) (1 + tan^{2} (x))^{10} tan^{21} (x) d x

Apply u-substitution

: \int u^{21} (1 + u^{2})^{10} d u

= \int u^{21} (1 + u^{2})^{10} d u

Expand

u^{21} (1 + u^{2})^{10} : u^{21} + 10 u^{23} + 45 u^{25} + 120 u^{27} + 210 u^{29} + 252 u^{31} + 210 u^{33} + 120 u^{35} + 45 u^{37} + 10 u^{39} + u^{41}

= \int u^{21} + 10 u^{23} + 45 u^{25} + 120 u^{27} + 210 u^{29} + 252 u^{31} + 210 u^{33} + 120 u^{35} + 45 u^{37} + 10 u^{39} + u^{41} 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 u^{21} d u + \int 10 u^{23} d u + \int 45 u^{25} d u + \int 120 u^{27} d u + \int 210 u^{29} d u + \int 252 u^{31} d u + \int 210 u^{33} d u + \int 120 u^{35} d u + \int 45 u^{37} d u + \int 10 u^{39} d u + \int u^{41} d u

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

\int 10 u^{23} d u = \frac{5 u ^{24}}{12}

\int 45 u^{25} d u = \frac{45 u ^{26}}{26}

\int 120 u^{27} d u = \frac{30 u ^{28}}{7}

\int 210 u^{29} d u = 7 u^{30}

\int 252 u^{31} d u = \frac{63 u ^{32}}{8}

\int 210 u^{33} d u = \frac{105 u ^{34}}{17}

\int 120 u^{35} d u = \frac{10 u ^{36}}{3}

\int 45 u^{37} d u = \frac{45 u ^{38}}{38}

\int 10 u^{39} d u = \frac{u ^{40}}{4}

\int u^{41} d u = \frac{u ^{42}}{42}

= \frac{u ^{22}}{22} + \frac{5 u ^{24}}{12} + \frac{45 u ^{26}}{26} + \frac{30 u ^{28}}{7} + 7 u^{30} + \frac{63 u ^{32}}{8} + \frac{105 u ^{34}}{17} + \frac{10 u ^{36}}{3} + \frac{45 u ^{38}}{38} + \frac{u ^{40}}{4} + \frac{u ^{42}}{42}

Substitute back

u = tan (x)

= \frac{tan ^{22} ( x )}{22} + \frac{5 tan ^{24} ( x )}{12} + \frac{45 tan ^{26} ( x )}{26} + \frac{30 tan ^{28} ( x )}{7} + 7 tan^{30} (x) + \frac{63 tan ^{32} ( x )}{8} + \frac{105 tan ^{34} ( x )}{17} + \frac{10 tan ^{36} ( x )}{3} + \frac{45 tan ^{38} ( x )}{38} + \frac{tan ^{40} ( x )}{4} + \frac{tan ^{42} ( x )}{42}

Add a constant to the solution

= \frac{tan ^{22} ( x )}{22} + \frac{5 tan ^{24} ( x )}{12} + \frac{45 tan ^{26} ( x )}{26} + \frac{30 tan ^{28} ( x )}{7} + 7 tan^{30} (x) + \frac{63 tan ^{32} ( x )}{8} + \frac{105 tan ^{34} ( x )}{17} + \frac{10 tan ^{36} ( x )}{3} + \frac{45 tan ^{38} ( x )}{38} + \frac{tan ^{40} ( x )}{4} + \frac{tan ^{42} ( x )}{42} + C

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What is the integral of tan^{21}(x)sec^{22}(x) ?
The integral of tan^{21}(x)sec^{22}(x) is