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受欢迎的微积分 >

integral of 28x^{-1/5}tan(x^{4/5})

解答

\int 28 x^{- \frac{1}{5}} tan (x^{\frac{4}{5}}) d x

解答

- 35 ln cos (x^{\frac{4}{5}}) + C

求解步骤

\int 28 x^{- \frac{1}{5}} tan (x^{\frac{4}{5}}) d x

提出常数:

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

= 28 \cdot \int x^{- \frac{1}{5}} tan (x^{\frac{4}{5}}) d x

使用换元积分法

= 28 \cdot \int \frac{5 tan ( u )}{4} d u

提出常数:

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

= 28 \cdot \frac{5}{4} \cdot \int tan (u) d u

使用三角恒等式改写

= 28 \cdot \frac{5}{4} \cdot \int \frac{sin ( u )}{cos ( u )} d u

使用换元积分法

= 28 \cdot \frac{5}{4} \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

= 28 \cdot \frac{5}{4} (- \int \frac{1}{v} d v)

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 28 \cdot \frac{5}{4} (- ln ∣ v ∣)

代回

= 28 \cdot \frac{5}{4} (- ln cos (x^{\frac{4}{5}}))

化简

28 \cdot \frac{5}{4} (- ln cos (x^{\frac{4}{5}})) : - 35 ln cos (x^{\frac{4}{5}})

= - 35 ln cos (x^{\frac{4}{5}})

解答补常数

= - 35 ln cos (x^{\frac{4}{5}}) + C

作图

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