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受欢迎的微积分 >

integral of tan^2(2/x) 1/(x^2)

解答

\int tan^{2} (\frac{2}{x}) \frac{1}{x ^{2}} d x

解答

\frac{1}{x} - \frac{1}{2} tan (\frac{2}{x}) + C

求解步骤

\int tan^{2} (\frac{2}{x}) \frac{1}{x ^{2}} d x

化简

tan^{2} (\frac{2}{x}) \frac{1}{x ^{2}} : \frac{tan ^{2} ( \frac{2}{x} )}{x ^{2}}

= \int \frac{tan ^{2} ( \frac{2}{x} )}{x ^{2}} d x

使用换元积分法

= \int - \frac{tan ^{2} ( u )}{2} d u

提出常数:

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

= - \frac{1}{2} \cdot \int tan^{2} (u) d u

使用三角恒等式改写

= - \frac{1}{2} \cdot \int - 1 + sec^{2} (u) d u

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}{2} (- \int 1 d u + \int sec^{2} (u) d u)

\int 1 d u = u

\int sec^{2} (u) d u = tan (u)

= - \frac{1}{2} (- u + tan (u))

u = \frac{2}{x}

代回

= - \frac{1}{2} (- \frac{2}{x} + tan (\frac{2}{x}))

化简

- \frac{1}{2} (- \frac{2}{x} + tan (\frac{2}{x})) : \frac{1}{x} - \frac{1}{2} tan (\frac{2}{x})

= \frac{1}{x} - \frac{1}{2} tan (\frac{2}{x})

解答补常数

= \frac{1}{x} - \frac{1}{2} tan (\frac{2}{x}) + C

作图

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