# 積分則

$\mathcal{L}\text{\hspace{0.17em}}\left\{{\int }_{0}^{t}f\left(t\right)dt\right\}=\frac{F\left(s\right)}{s}+\frac{{f}^{\left(-1\right)}\left(0\right)}{s}$

ただし，${f}^{\left(-1\right)}\left(0\right)={\int }_{-\infty }^{0}f\left(t\right)dt$

$\mathcal{L}\text{\hspace{0.17em}}\left\{\frac{1}{t}f\left(t\right)\right\}={\int }_{s}^{\infty }F\left(s\right)ds$

$\mathcal{L}\left\{{\int }_{0}^{t}{\int }_{0}^{t}{\int }_{0}^{t}\cdots {\int }_{0}^{t}f\left(t\right){\left(dt\right)}^{n}\right\}$$=\frac{F\left(s\right)}{{s}^{n}}+\frac{{f}^{\left(-1\right)}\left(0\right)}{{s}^{n}}+\frac{{f}^{\left(-2\right)}\left(0\right)}{{s}^{n-1}}+$$\cdots +\frac{{f}^{\left(-n\right)}\left(0\right)}{s}$

ただし

${f}^{\left(-n\right)}\left(0\right)={\int }_{-\text{∞}}^{0}{\int }_{-\text{∞}}^{0}{\int }_{-\text{∞}}^{0}$$\cdots {\int }_{-\text{∞}}^{0}f\left(t\right){\left(dt\right)}^{n}$

$\mathcal{L}\left\{\frac{1}{{t}^{n}}f\left(t\right)\right\}={\int }_{s}^{\text{∞}}{\int }_{s}^{\text{∞}}{\int }_{s}^{\text{∞}}$${\cdots \int }_{s}^{\text{∞}}{F\left(s\right)\left(ds\right)}^{n}$

## ■証明

$\mathcal{L}\text{\hspace{0.17em}}\left\{{\int }_{0}^{t}f\left(t\right)dt\right\}={\int }_{0}^{\infty }{e}^{-st}\left\{{\int }_{0}^{t}f\left(t\right)dt\right\}dt$

$={\int }_{0}^{\infty }{\left(-\frac{1}{s}{e}^{-st}\right)}^{\text{'}}\left\{{\int }_{0}^{t}f\left(t\right)dt\right\}dt$

$={\left[-\frac{1}{s}{e}^{-st}{\int }_{0}^{t}f\left(t\right)dt\right]}_{0}^{\text{∞}}$$-{\int }_{0}^{\text{∞}}\left(-\frac{1}{s}{e}^{-st}\right)\frac{d}{dt}\left\{{\int }_{0}^{t}f\left(t\right)dt\right\}dt$

$={\left[-\frac{1}{s}{e}^{-st}{\int }_{0}^{t}f\left(t\right)dt\right]}_{0}^{\text{∞}}$$+\frac{1}{s}{\int }_{0}^{\text{∞}}{e}^{-st}f\left(t\right)dt$

$=\frac{1}{s}{\int }_{0}^{\infty }{e}^{-st}f\left(t\right)dt$

$=\frac{F\left(s\right)}{s}$

## ■証明

${\int }_{s}^{\infty }F\left(s\right)ds$$={\int }_{s}^{\infty }{\int }_{0}^{\infty }{e}^{-st}f\left(t\right)dtds$

$={\int }_{0}^{\infty }\left\{{\int }_{s}^{\infty }{e}^{-st}f\left(t\right)ds\right\}dt$

$={\int }_{0}^{\infty }f\left(t\right){\int }_{s}^{\infty }{e}^{-st}dsdt$

$={\int }_{0}^{\infty }f\left(t\right){\left[-\frac{1}{t}{e}^{-st}\right]}_{s}^{\infty }dt$

$={\int }_{0}^{\infty }f\left(t\right)\left\{0-\left(-\frac{1}{t}{e}^{-st}\right)\right\}dt$

$={\int }_{0}^{\infty }{e}^{-st}\left\{\frac{f\left(t\right)}{t}\right\}dt$

$=\mathcal{L}\text{\hspace{0.17em}}\left\{\frac{1}{t}f\left(t\right)\right\}$

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最終更新日： 2023年6月6日