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受欢迎的微积分 >

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

解答

\int \frac{1}{5 ^{4 x}} d x

解答

- \frac{62 5 ^{- x}}{4 ln ( 5 )} + C

求解步骤

\int \frac{1}{5 ^{4 x}} d x

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{5 ^{4 x}} = 5^{- 4 x}

= \int 5^{- 4 x} d x

化简

5^{- 4 x} : 62 5^{- x}

= \int 62 5^{- x} d x

使用换元积分法

= \int - 62 5^{u} d u

提出常数:

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

= - \int 62 5^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= - \frac{62 5 ^{u}}{ln ( 625 )}

u = - x

代回

= - \frac{62 5 ^{- x}}{ln ( 625 )}

化简

ln (625) : 4 ln (5)

= - \frac{62 5 ^{- x}}{4 ln ( 5 )}

解答补常数

= - \frac{62 5 ^{- x}}{4 ln ( 5 )} + C

作图

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