integral from pi/(12) to pi/6 of cot(3x)+csc(3x)

解答

\int_{\frac{π}{12}}^{\frac{π}{6}} cot (3 x) + csc (3 x) d x

\frac{1}{6} ln (2) - \frac{1}{3} ln (2 - 1)

十进制

0.40931 \dots

求解步骤

\int_{\frac{π}{12}}^{\frac{π}{6}} cot (3 x) + csc (3 x) d x

使用三角恒等式改写

= \int_{\frac{π}{12}}^{\frac{π}{6}} \frac{1}{tan ( 3 x )} + csc (3 x) d x

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

= \int_{\frac{π}{12}}^{\frac{π}{6}} \frac{1}{tan ( 3 x )} d x + \int_{\frac{π}{12}}^{\frac{π}{6}} csc (3 x) d x

\int_{\frac{π}{12}}^{\frac{π}{6}} \frac{1}{tan ( 3 x )} d x = \frac{1}{6} ln (2)

\int_{\frac{π}{12}}^{\frac{π}{6}} csc (3 x) d x = - \frac{1}{3} ln (2 - 1)

= \frac{1}{6} ln (2) - \frac{1}{3} ln (2 - 1)

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