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受欢迎的微积分 >

integral of (cos^5(3x))/(sin^{14)(3x)}

解答

\int \frac{cos ^{5} ( 3 x )}{sin ^{14} ( 3 x )} d x

解答

\frac{1}{3} (- \frac{1}{13} csc^{13} (3 x) + \frac{2}{11} csc^{11} (3 x) - \frac{1}{9} csc^{9} (3 x)) + C

求解步骤

\int \frac{cos ^{5} ( 3 x )}{sin ^{14} ( 3 x )} d x

使用换元积分法

= \int \frac{cos ^{5} ( u )}{3 sin ^{14} ( u )} d u

提出常数:

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

= \frac{1}{3} \cdot \int \frac{cos ^{5} ( u )}{sin ^{14} ( u )} d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) ^{2} cos ( u )}{sin ^{14} ( u )} d u

使用换元积分法

= \frac{1}{3} \cdot \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{14}} d v

乘开

\frac{( 1 - v ^{2} ) ^{2}}{v ^{14}} : \frac{1}{v ^{14}} - \frac{2}{v ^{12}} + \frac{1}{v ^{10}}

= \frac{1}{3} \cdot \int \frac{1}{v ^{14}} - \frac{2}{v ^{12}} + \frac{1}{v ^{10}} d v

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{1}{3} (\int \frac{1}{v ^{14}} d v - \int \frac{2}{v ^{12}} d v + \int \frac{1}{v ^{10}} d v)

\int \frac{1}{v ^{14}} d v = - \frac{1}{13 v ^{13}}

\int \frac{2}{v ^{12}} d v = - \frac{2}{11 v ^{11}}

\int \frac{1}{v ^{10}} d v = - \frac{1}{9 v ^{9}}

= \frac{1}{3} (- \frac{1}{13 v ^{13}} - (- \frac{2}{11 v ^{11}}) - \frac{1}{9 v ^{9}})

代回

= \frac{1}{3} (- \frac{1}{13 sin ^{13} ( 3 x )} - (- \frac{2}{11 sin ^{11} ( 3 x )}) - \frac{1}{9 sin ^{9} ( 3 x )})

化简

\frac{1}{3} (- \frac{1}{13 sin ^{13} ( 3 x )} - (- \frac{2}{11 sin ^{11} ( 3 x )}) - \frac{1}{9 sin ^{9} ( 3 x )}) : \frac{1}{3} (- \frac{1}{13} csc^{13} (3 x) + \frac{2}{11} csc^{11} (3 x) - \frac{1}{9} csc^{9} (3 x))

= \frac{1}{3} (- \frac{1}{13} csc^{13} (3 x) + \frac{2}{11} csc^{11} (3 x) - \frac{1}{9} csc^{9} (3 x))

解答补常数

= \frac{1}{3} (- \frac{1}{13} csc^{13} (3 x) + \frac{2}{11} csc^{11} (3 x) - \frac{1}{9} csc^{9} (3 x)) + C

作图

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