日: 2013年12月11日

Integrals

Integrals

If dy/dx = f(x), then it’s called that y an indefinite integral of f(x) and denoted as

\displaystyle \int f(x)dx

If f(x) = \frac{d}{dx}F(x), then

\displaystyle \int_a^b f(x)dx = F(b) - F(a)

Integral formulas

In the following u, v represent functions of x while a, b, c, p represent constants.

\displaystyle \int (u \pm v)dx = \int u dx \pm \int v dx\\\vspace{0.2 in} \int cu dx = c\int u dx \\\vspace{0.2 in} \int u\left(\frac{dv}{dx}\right) = uv - \int v \left(\frac{du}{dx}\right)dx \\\vspace{0.2 in} \int u dv = uv - \int v du

This is called integration by parts.

\displaystyle \int F(u(x))dx = \int F(w)\frac{dw}{dw/dx}

where w = u(x) and w’ = dw/dx expressed as a function of w. This is called integration by substitution or tranformation.

\displaystyle \int u^p du = \frac{u^{p+1}}{p+1},\ p \neq -1\\\vspace{0.2 in} \int u^{-1}du = \int \frac{du}{u} = \ln u\\\vspace{0.2 in} \int a^u du = \frac{a^u}{\ln a},\ a \neq 0,\ 1\\\vspace{0.2 in} \int e^u du = e^u

\displaystyle \int \sin u\ du = -\cos{u}\\\vspace{0.2 in} \int \cos u\ du = \sin{u}\\\vspace{0.2 in} \int \tan u\ du = -\ln \cos{u}\\\vspace{0.2 in} \int e^{au}\sin{bu}\ du = \frac{e^{au}(a\ \sin{bu}- b\ \cos{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int e^{au}\cos{bu}\ du = \frac{e^{au}(a\ \cos{bu}+ b\ \sin{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{u^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 - a^2}} = \ln(u + \sqrt{u^2 - a^2})\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 + a^2}} = \ln(u + \sqrt{u^2 + a^2})

積分

積分

　ある関数について dy/dx = f(x) である時，y は次のように表現されます．

\displaystyle \int f(x)dx

　f(x) = \frac{d}{dx}F(x) の時，積分は以下のように定義されます．

\displaystyle \int_a^b f(x)dx = F(b) - F(a)

積分公式

　関数 u, v および定数 a, b, c, p について下記が成り立ちます．

\displaystyle \int (u \pm v)dx = \int u dx \pm \int v dx\\\vspace{0.2 in} \int cu dx = c\int u dx \\\vspace{0.2 in} \int u\left(\frac{dv}{dx}\right) = uv - \int v \left(\frac{du}{dx}\right)dx \\\vspace{0.2 in} \int u dv = uv - \int v du \\\vspace{0.2 in} \int F(u(x))dx = \int F(w)\frac{dw}{dw/dx},\ w = u(x) \displaystyle \int u^p du = \frac{u^{p+1}}{p+1},\ p \neq -1\\\vspace{0.2 in} \int u^{-1}du = \int \frac{du}{u} = \ln u\\\vspace{0.2 in} \int a^u du = \frac{a^u}{\ln a},\ a \neq 0,\ 1\\\vspace{0.2 in} \int e^u du = e^u \displaystyle \int \sin u\ du = -\cos{u}\\\vspace{0.2 in} \int \cos u\ du = \sin{u}\\\vspace{0.2 in} \int \tan u\ du = -\ln \cos{u}\\\vspace{0.2 in} \int e^{au}\sin{bu}\ du = \frac{e^{au}(a\ \sin{bu}- b\ \cos{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int e^{au}\cos{bu}\ du = \frac{e^{au}(a\ \cos{bu}+ b\ \sin{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{u^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 - a^2}} = \ln(u + \sqrt{u^2 - a^2})\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 + a^2}} = \ln(u + \sqrt{u^2 + a^2})