What is the integral of tan^{11}(x/4)csc^6(x/4) ?

The integral of tan^{11}(x/4)csc^6(x/4) is 1/15 tan^6(x/4)(6tan^4(x/4)+15tan^2(x/4)+10)+C

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integral of tan^{11}(x/4)csc^6(x/4)

\int tan^{11} (\frac{x}{4}) csc^{6} (\frac{x}{4}) d x

\frac{1}{15} tan^{6} (\frac{x}{4}) (6 tan^{4} (\frac{x}{4}) + 15 tan^{2} (\frac{x}{4}) + 10) + C

Solution steps

\int tan^{11} (\frac{x}{4}) csc^{6} (\frac{x}{4}) d x

Apply u-substitution

: \int \frac{4 ( w ^{\frac{2}{11}} + 1 ) ^{2}}{11 w ^{\frac{5}{11}}} d w

= \int \frac{4 ( w ^{\frac{2}{11}} + 1 ) ^{2}}{11 w ^{\frac{5}{11}}} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{4}{11} \cdot \int \frac{( w ^{\frac{2}{11}} + 1 ) ^{2}}{w ^{\frac{5}{11}}} d w

Expand

\frac{( w ^{\frac{2}{11}} + 1 ) ^{2}}{w ^{\frac{5}{11}}} : \frac{1}{w ^{\frac{1}{11}}} + \frac{2}{w ^{\frac{3}{11}}} + \frac{1}{w ^{\frac{5}{11}}}

= \frac{4}{11} \cdot \int \frac{1}{w ^{\frac{1}{11}}} + \frac{2}{w ^{\frac{3}{11}}} + \frac{1}{w ^{\frac{5}{11}}} d w

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{4}{11} (\int \frac{1}{w ^{\frac{1}{11}}} d w + \int \frac{2}{w ^{\frac{3}{11}}} d w + \int \frac{1}{w ^{\frac{5}{11}}} d w)

\int \frac{1}{w ^{\frac{1}{11}}} d w = \frac{11}{10} w^{\frac{10}{11}}

\int \frac{2}{w ^{\frac{3}{11}}} d w = \frac{11}{4} w^{\frac{8}{11}}

\int \frac{1}{w ^{\frac{5}{11}}} d w = \frac{11}{6} w^{\frac{6}{11}}

= \frac{4}{11} (\frac{11}{10} w^{\frac{10}{11}} + \frac{11}{4} w^{\frac{8}{11}} + \frac{11}{6} w^{\frac{6}{11}})

Substitute back

w = tan^{11} (\frac{x}{4})

= \frac{4}{11} (\frac{11}{10} (tan^{11} (\frac{x}{4}))^{\frac{10}{11}} + \frac{11}{4} (tan^{11} (\frac{x}{4}))^{\frac{8}{11}} + \frac{11}{6} (tan^{11} (\frac{x}{4}))^{\frac{6}{11}})

Simplify

\frac{4}{11} (\frac{11}{10} (tan^{11} (\frac{x}{4}))^{\frac{10}{11}} + \frac{11}{4} (tan^{11} (\frac{x}{4}))^{\frac{8}{11}} + \frac{11}{6} (tan^{11} (\frac{x}{4}))^{\frac{6}{11}}) : \frac{1}{15} tan^{6} (\frac{x}{4}) (6 tan^{4} (\frac{x}{4}) + 15 tan^{2} (\frac{x}{4}) + 10)

= \frac{1}{15} tan^{6} (\frac{x}{4}) (6 tan^{4} (\frac{x}{4}) + 15 tan^{2} (\frac{x}{4}) + 10)

Add a constant to the solution

= \frac{1}{15} tan^{6} (\frac{x}{4}) (6 tan^{4} (\frac{x}{4}) + 15 tan^{2} (\frac{x}{4}) + 10) + C

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What is the integral of tan^{11}(x/4)csc^6(x/4) ?
The integral of tan^{11}(x/4)csc^6(x/4) is 1/15 tan^6(x/4)(6tan^4(x/4)+15tan^2(x/4)+10)+C