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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

提出常数:

\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

乘开

\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

使用积分加法定则:

\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}})

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

代回

= \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}})

化简

\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)

解答补常数

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

作图

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