integral of (e^x-e^5)(e^x+e^{-3})

解答

\int (e^{x} - e^{5}) (e^{x} + e^{- 3}) d x

\frac{1}{2} e^{2 x} + e^{x - 3} - e^{5 + x} - e^{2} x + C

求解步骤

\int (e^{x} - e^{5}) (e^{x} + e^{- 3}) d x

乘开

(e^{x} - e^{5}) (e^{x} + e^{- 3}) : e^{2 x} + e^{x - 3} - e^{5 + x} - e^{2}

= \int e^{2 x} + e^{x - 3} - e^{5 + x} - e^{2} d x

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 e^{2 x} d x + \int e^{x - 3} d x - \int e^{5 + x} d x - \int e^{2} d x

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

\int e^{x - 3} d x = e^{x - 3}

\int e^{5 + x} d x = e^{5 + x}

\int e^{2} d x = e^{2} x

= \frac{1}{2} e^{2 x} + e^{x - 3} - e^{5 + x} - e^{2} x

解答补常数

= \frac{1}{2} e^{2 x} + e^{x - 3} - e^{5 + x} - e^{2} x + C

a re a x = 7 y^{2}, x = 2 + 5 y^{2}

\int 200 e^{0.02 x} d x

\int \frac{x ^{2}}{16 - x ^{6}} d x

\frac{d}{d x} (x sec^{2} (x))

\frac{d y}{d x} y = 11 x