integral from 0 to a of e^{x/a}-e^{-x/a}

解

\int_{0}^{a} e^{\frac{x}{a}} - e^{- \frac{x}{a}} d x

e a + \frac{- 2 e + 1}{e} a

解答ステップ

\int_{0}^{a} e^{\frac{x}{a}} - e^{- \frac{x}{a}} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{a} e^{\frac{x}{a}} d x - \int_{0}^{a} e^{- \frac{x}{a}} d x

\int_{0}^{a} e^{\frac{x}{a}} d x = a (e - 1)

\int_{0}^{a} e^{- \frac{x}{a}} d x = a (1 - \frac{1}{e})

= a (e - 1) - a (1 - \frac{1}{e})

簡素化

a (e - 1) - a (1 - \frac{1}{e}) : e a + \frac{- 2 e + 1}{e} a

= e a + \frac{- 2 e + 1}{e} a