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受欢迎的微积分 >

integral of e^{-5x}tan(e^{-5x})

解答

\int e^{- 5 x} tan (e^{- 5 x}) d x

解答

\frac{1}{5} ln cos (e^{- 5 x}) + C

求解步骤

\int e^{- 5 x} tan (e^{- 5 x}) d x

使用换元积分法

= \int - \frac{tan ( u )}{5} d u

提出常数:

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

= - \frac{1}{5} \cdot \int tan (u) d u

使用三角恒等式改写

= - \frac{1}{5} \cdot \int \frac{sin ( u )}{cos ( u )} d u

使用换元积分法

= - \frac{1}{5} \cdot \int - \frac{1}{v} d v

提出常数:

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

= - \frac{1}{5} (- \int \frac{1}{v} d v)

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{5} (- ln ∣ v ∣)

代回

= - \frac{1}{5} (- ln cos (e^{- 5 x}))

化简

= \frac{1}{5} ln cos (e^{- 5 x})

解答补常数

= \frac{1}{5} ln cos (e^{- 5 x}) + C

作图

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