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Definite integrals

Let F:I→RF:I\to \mathbb{R} be an antiderivative of the continuous function f:I→Rf:I\to \mathbb{R}. The definite integral of f from a to b is the real number:

∫abf(x)dx=F(b)−F(a)\displaystyle \int\limits_{a}^{b}{f(x)}dx=F(b)-F(a) (the Leibniz–Newton formula)

Properties:

  • Linearity of the integral:

∀λ,μ∈R,∫ab(λf(x)+μg(x))dx=λ∫abf(x)dx+μ∫abg(x)dx.\displaystyle \forall \lambda ,\mu \in \mathbb{R},\int\limits_{a}^{b}{(\lambda f(x)+\mu g(x))dx=\lambda \int\limits_{a}^{b}{f(x)dx+\mu \int\limits_{a}^{b}{g(x)dx.}}}

  • Additivity of the integral: ∀c∈I,∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx.\displaystyle \forall c\in I, \int\limits_{a}^{b}{f(x)dx}=\int\limits_{a}^{c}{f(x)dx+\int\limits_{c}^{b}{f(x)dx.}}

  • The mean value property of the integral: if f≥0f\ge 0 on [a,b],∫zbf(x)dx≥0\displaystyle [a,b],\int\limits_{z}^{b}{f(x)dx\ge 0}

  • Monotonicity of the integral: if f≤gf\le g on [a,b],∫abf(x)dx≤∫abg(x)dx.\displaystyle [a,b], \int\limits_{a}^{b}{f(x)dx}\le \int\limits_{a}^{b}{g(x)dx}.

Geometric interpretation of the definite integral

Let a<b be real numbers and f:[a,b]→Rf:[a,b]\to \mathbb{R} a positive continuous function. The region under the graph of f is Γf={(x,y)∈R2∣a≤x≤b,0≤y≤f(x)}.{{\Gamma }_{f}}=\{(x,y)\in {{\mathbb{R}}^{2}}\left| a\le x\le b, 0\le y\le f(x)\}. \right.

The Riemann sum (or integral sum) associated with f, Δ\Delta and ξ\xi is the real number σ=∑i=1nf(ξ)⋅(xi−xi−1)\displaystyle \sigma = \sum_{i=1}^{n} f(\xi) \cdot (x_i - x_{i-1}) where ξ=(ξ1,ξ2,...,ξn)=(ξi)1≤i≤n\xi =({{\xi }_{1}},{{\xi }_{2}},...,{{\xi }_{n}})={{({{\xi }_{i}})}_{1\le i\le n}} with ξi∈[xi−1,xi]{{\xi }_{i}}\in [{{x}_{i-1}},{{x}_{i}}], i∈{1,2,...,n}i\in \{1,2,...,n\} being the system of intermediate points associated with the partition: Δ=(a=x0<x1<...<xn−1=b)\Delta =(a={{x}_{0}}<{{x}_{1}}<...<{{x}_{n-1}}=b).

The area under the graph of a positive continuous function f:[a,b]→Rf:[a,b]\to \mathbb{R} is: area (Γf)=∫abf(x)dx\displaystyle ({{\Gamma }_{f}})=\int\limits_{a}^{b}{f(x)dx}.

The Riemann integral of piecewise continuous functions

A function f:[a,b]→Rf:[a,b]\to \mathbb{R} is called piecewise continuous if it has at most a finite, non-zero number of points of discontinuity, and these are discontinuities of the first kind.

Integrating piecewise continuous functions. A piecewise continuous function f:[a,b]→Rf:[a,b]\to \mathbb{R} is integrable on [a,b][a,b] and ∫abf(x)dx=∑i=1p∫ci−1cifi(x)dx\displaystyle \int\limits_{a}^{b}{f(x)dx}=\sum\limits_{i=1}^{p}{\int\limits_{{{c}_{i-1}}}^{{{c}_{i}}}{{{f}_{i}}(x)dx}}, where fi:[ci−1,ci]→R,i∈{1,...,p},{{f}_{i}}:[{{c}_{i-1}},{{c}_{i}}]\to \mathbb{R},i\in \{1,...,p\}, are the continuous functions associated with f.

Methods for computing definite integrals

Integration by parts. Suppose the functions f,g:I→Rf,g:I\to \mathbb{R} are differentiable, with continuous derivatives f′,g′:I→R{{f}^{'}},{{g}^{'}}:I\to \mathbb{R}. Let a,b∈Ia,b\in I be two numbers. Then:

∫abf(x)g′(x)dx=f(x)g(x)∣ab−∫abg(x)f′(x)dx.\displaystyle \int\limits_{a}^{b}{f(x)g'(x)dx=\left. f(x)g(x) \right|_{a}^{b}-\int\limits_{a}^{b}{g(x)f'(x)dx.}}

Change of variable. Suppose the function φ:J→I\varphi :J\to I is differentiable with continuous derivative, and the function F:I→RF:I\to \mathbb{R} is continuous. Let α,β∈J\alpha ,\beta \in J be two numbers. Then: ∫αβf(φ(t))⋅φ′(t)dt=∫φ(α)φ(β)f(x)dx.\displaystyle \int\limits_{\alpha }^{\beta }{f(\varphi (t))\cdot \varphi '(t)dt=\int\limits_{\varphi (\alpha )}^{\varphi (\beta )}{f(x)dx.}} One changes both variable and symbol, φ(t)=x\varphi (t)=x and φ′(t)dt=dx,t∈J\varphi '(t)dt=dx, t\in J and x∈Ix\in I.

Further properties:

∫aaf(x)dx=0\displaystyle \int\limits_{a}^{a}{f(x)}dx=0

∫abf(x)dx=−∫baf(x)dx\displaystyle \int\limits_{a}^{b}{f(x)}dx=-\int\limits_{b}^{a}{f(x)dx}

∫ab∑i=1nki⋅fi(x) dx=∑i=1nki∫abfi(x) dx,  k∈R\displaystyle \int_a^b \sum_{i=1}^{n} k_i \cdot f_i(x)\,dx = \sum_{i=1}^{n} k_i \int_a^b f_i(x)\,dx,\; k \in \mathbb{R}

The mean value property of the integral

For the function ff, there exists a number c∈(a,b)c\in (a,b) such that

∫abf(x)dx=f(c)⋅(b−a).\displaystyle \int\limits_{a}^{b}{f(x)}dx=f(c)\cdot (b-a).

The mean value inequality

If m,M∈Rm,M\in \mathbb{R} are such that m≤f(x)≤Mm\le f(x)\le M, ∀x∈[a,b]\forall x\in [a,b], then

∫abmdx≤∫abf(x)dx≤∫abMdx⇒m(b−a)≤∫abf(x)dx≤M(b−a).\displaystyle \int\limits_{a}^{b}{m}dx\le \int\limits_{a}^{b}{f(x)}dx\le \int\limits_{a}^{b}{M}dx\Rightarrow m(b-a)\le \int\limits_{a}^{b}{f(x)}dx\le M(b-a).

(∫axf(t)dt)′=f(x)\displaystyle {{\left( \int\limits_{a}^{x}{f(t)}dt \right)}^{\prime }}=f(x)

| | | | ∫0∞dxa+bx2=π2ab\displaystyle \int\limits_{0}^{\infty }{\frac{dx}{a+b{{x}^{2}}}}=\frac{\pi }{2\sqrt{ab}} | ∫0πln⁡∣cos⁡(x)∣dx=∫0πln⁡(sin⁡(x))dx=−πln⁡(2)\displaystyle \int\limits_{0}^{\pi }{\ln \left| \cos (x) \right|}dx=\int\limits_{0}^{\pi }{\ln (\sin (x))}dx=-\pi \ln (2) | | ∫0∞sin⁡(x)xdx=π2\displaystyle \int\limits_{0}^{\infty }{\frac{\sin (x)}{x}}dx=\frac{\pi }{2} | ∫0∞tg(x)xdx=π2\displaystyle \int\limits_{0}^{\infty }{\frac{tg(x)}{x}}dx=\frac{\pi }{2} | | ∫0∞e−xdx=1\displaystyle \int\limits_{0}^{\infty }{{{e}^{-x}}}dx=1 | ∫0∞e−x2dx=π2\displaystyle \int\limits_{0}^{\infty }{{{e}^{-{{x}^{2}}}}}dx=\frac{\sqrt{\pi }}{2} | | ∫02πx⋅sin⁡(x)dx=−2π\displaystyle \int\limits_{0}^{2\pi }{x\cdot \sin (x)}dx=-2\pi | ∫0πsin⁡(mx)⋅sin⁡(nx)dx=∫0πcos⁡(mx)⋅cos⁡(nx)dx={0, dac  ⁣ ⁣a˘ ⁣ ⁣ m≠nπ2, dac  ⁣ ⁣a˘ ⁣ ⁣ m=n m,n∈Z\displaystyle \begin{aligned} & \int\limits_{0}^{\pi }{\sin (mx)\cdot \sin (nx)}dx= \\ & \int\limits_{0}^{\pi }{\cos (mx)\cdot \cos (nx)}dx=\left\{ \begin{aligned} & 0,\text{ dac }\!\!\breve{\mathrm{a}}\!\!\text{ m}\ne \text{n} \\ & \frac{\pi }{2},\text{ dac }\!\!\breve{\mathrm{a}}\!\!\text{ m=n} \\ \end{aligned} \right.\text{ }m,n\in \mathbb{Z} \\ \end{aligned} |