integral from-1 to 0 of (e^{1/θ})/(θ^3)

解

\int_{- 1}^{0} \frac{e ^{\frac{1}{θ}}}{θ ^{3}} d θ

- \frac{2}{e}

十進法表記

- 0.73575 \dots

解答ステップ

\int_{- 1}^{0} \frac{e ^{\frac{1}{θ}}}{θ ^{3}} d θ

部分積分を適用する

= [\frac{e ^{\frac{1}{θ}}}{θ ^{2}} - \int - \frac{e ^{\frac{1}{θ}} ( 3 θ + 1 )}{θ ^{4}} d θ]_{- 1}^{0}

\int - \frac{e ^{\frac{1}{θ}} ( 3 θ + 1 )}{θ ^{4}} d θ = \frac{3 e ^{\frac{1}{θ}}}{θ} - e^{\frac{1}{θ}} + \frac{e ^{\frac{1}{θ}}}{θ ^{2}} - \frac{2 e ^{\frac{1}{θ}}}{θ}

= [\frac{e ^{\frac{1}{θ}}}{θ ^{2}} - (\frac{3 e ^{\frac{1}{θ}}}{θ} - e^{\frac{1}{θ}} + \frac{e ^{\frac{1}{θ}}}{θ ^{2}} - \frac{2 e ^{\frac{1}{θ}}}{θ})]_{- 1}^{0}

簡素化

[\frac{e ^{\frac{1}{θ}}}{θ ^{2}} - (\frac{3 e ^{\frac{1}{θ}}}{θ} - e^{\frac{1}{θ}} + \frac{e ^{\frac{1}{θ}}}{θ ^{2}} - \frac{2 e ^{\frac{1}{θ}}}{θ})]_{- 1}^{0} : [\frac{- e ^{\frac{1}{θ}} + e ^{\frac{1}{θ}} θ}{θ}]_{- 1}^{0}

= [\frac{- e ^{\frac{1}{θ}} + e ^{\frac{1}{θ}} θ}{θ}]_{- 1}^{0}

限度を計算する:

- \frac{2}{e}

= - \frac{2}{e}