积分 ((e^{-t}-4))/(e^{-t)}

解答

积分

\frac{( e ^{- t} - 4 )}{e ^{- t}}

t - 4 e^{t} + C

求解步骤

\int \frac{e ^{- t} - 4}{e ^{- t}} d t

化简

\frac{e ^{- t} - 4}{e ^{- t}} : e^{t} (e^{- t} - 4)

= \int e^{t} (e^{- t} - 4) d t

乘开

e^{t} (e^{- t} - 4) : 1 - 4 e^{t}

= \int 1 - 4 e^{t} d t

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 1 d t - \int 4 e^{t} d t

\int 1 d t = t

\int 4 e^{t} d t = 4 e^{t}

= t - 4 e^{t}

解答补常数

= t - 4 e^{t} + C

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