Integration | Improve Society

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1 Integrals
2 Integral formulas

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.
$\displaystyle \int u^p du = \frac{u^{p+1}}{p+1},\ p \neq -1\\\vspace{0.2 in} \int u^{-1}du = \int \frac{du}{u} = \ln u\\\vspace{0.2 in} \int a^u du = \frac{a^u}{\ln a},\ a \neq 0,\ 1\\\vspace{0.2 in} \int e^u du = e^u$ $\displaystyle \int \sin u\ du = -\cos{u}\\\vspace{0.2 in} \int \cos u\ du = \sin{u}\\\vspace{0.2 in} \int \tan u\ du = -\ln \cos{u}\\\vspace{0.2 in} \int e^{au}\sin{bu}\ du = \frac{e^{au}(a\ \sin{bu}- b\ \cos{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int e^{au}\cos{bu}\ du = \frac{e^{au}(a\ \cos{bu}+ b\ \sin{bu})}{a^2 + b^2}\\\vspace{0.2 in} \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{u^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{u}{a}\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 - a^2}} = \ln(u + \sqrt{u^2 - a^2})\\\vspace{0.2 in} \int \frac{du}{\sqrt{u^2 + a^2}} = \ln(u + \sqrt{u^2 + a^2})$