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Integral Calculus
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Mugees Ul Kaisar (0)

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what is integration ? explain
    
mohamed arafath (20)

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integration means ... addition (+)
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sathyaram (150)

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  Integration of a function is familiar from introductory calculus courses, where integration is usually introduced in two ways. The first is as the inverse process to differentiation, as the so-called anti-derivative of a function. Thus if we have 

 

\begin{displaymath}\frac{df}{dx} = g(x)\end{displaymath}

 

 

then we invert this to say that f(x) is the integral of g(x). While this is not a particularly satisfying definition of integration, it does reflect the usual methods we use to evaluate integrals in closed form, when this is possible. As a very simple example, one simply says that the integral of $\cos(\theta)$ is $\sin(\theta)$, because the derivative of $\sin(\theta)$ with respect to $\theta$ is $\cos(\theta)$. Of course, there is an arbitrary additive constant to be added to the integral described in this way. In the notation of integral calculus, we would write: 

 

\begin{displaymath}\int^x g(t) dt = f(x) + C,\end{displaymath}

 

 

where C is the arbitrary constant with respect to xg(t) is called the integrand, and t is the variable of integration. It will be important to note that t is a ``dummy'' variable of integration, and that the integral itself is not a function of this ``dummy'' variable, but of the limiting values it takes. Consider the rather simple function: 

 

g(x) = A x2 + B x + C,

 

 

where AB, and C are constant with respect of x. To find the purely symbolic integral of g(x), we would evaluate: 

 

\begin{displaymath}\int^x (A t^2 + B t + C) dt = \frac{A}{3} x^3 + \frac{B}{2} x^2 + C x + D,\end{displaymath}

 

 

where D is a constant of integration. Integrals in this symbolic form are called indefinite integrals, since they do not yield any definite result until evaluated between actual limits, so that the constant of integration, D in this example, takes on a specific value. Thus a definite integral obtainable from the indefinite form would be: 

 

\begin{displaymath}F(\alpha,\beta) = \int_{\alpha}^{\beta} (A t^2 + B t + C) dt\end{displaymath}

 

 

which would yield the value: 

 

\begin{displaymath}F(\alpha,\beta) = \left [ \frac{A}{3} x^3 + \frac{B}{2} x^2 + C x \right ]_{x = \alpha}^{x = \beta}.\end{displaymath}

 

 

The [ ... ] expression means to evaluate the contents for the upper limit ($\beta$ in this example) and subtract from it the value of the contents evaluated at the lower limit ($\alpha$ in this example). Thus the quantity $F(\alpha,\beta)$ is not a function of xat all, but definitely a function of its two arguments, $\alpha$ and $\beta$.

In many applications of calculus to problems that arise in science and engineering, the integrals that are involved in calculating the relevant quantities are not easily evaluated in symbolic form. By this we mean that it is not easily possible to find the appropriate function, whose derivative is the integrand of the integral.

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sathyaram (150)

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 An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."

The Riemann integral of the function f(x) over x from a to b is written

 int_a^bf(x)dx.
(1)

Note that if f(x)=1, the integral is written simply

 int_a^bdx
(2)

as opposed to int_a^b1dx.

Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The process of computing an integral is called integration (a more archaic term for integration is quadrature), and the approximate computation of an integral is termed numerical integration.

There are two classes of (Riemann) integrals: definite integrals such as (1), which have upper and lower limits, and indefinite integrals, such as

 intf(x)dx,
(3)

which are written without limits. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if F(x)is the indefinite integral for f(x), then

 int_a^bf(x)dx=F(b)-F(a).
(4)

Since the derivative of a constant is zero, indefinite integrals are defined only up to an arbitrary constant of integration C, i.e.,

 intf(x)dx=F(x)+C.
(5)

Wolfram Research maintains a web site http://integrals.wolfram.com/ that can find the indefinite integral of many common (and not so common) functions.

Differentiating integrals leads to some useful and powerful identities. For instance, if f(x) is continuous, then

 d/(dx)int_a^xf(x^')dx^'=f(x),
(6)

which is the first fundamental theorem of calculus. Other derivative-integral identities include

 d/(dx)int_x^bf(x^')dx^'=-f(x),
(7)

the Leibniz integral rule

 d/(dx)int_a^bf(x,t)dt=int_a^bpartial/(partialx)f(x,t)dt
(8)

(Kaplan 1992, p. 275), its generalization

 d/(dx)int_(u(x))^(v(x))f(x,t)dt=v^'(x)f(x,v(x))-u^'(x)f(x,u(x))+int_(u(x))^(v(x))partial/(partialx)f(x,t)dt
(9)

(Kaplan 1992, p. 258), and

 d/(dx)int_a^xf(x,t)dt=1/(x-a)int_a^x[(x-a)partial/(partialx)f(x,t)+(t-a)partial/(partialt)f(x,t)+f(x,t)]dt,
(10)

as can be seen by applying (9) on the left side of (10) and using partial integration.

Other integral identities include

 int_0^xdt_nint_0^(t_n)dt_(n-1)...int_0^(t_3)dt_2int_0^(t_2)f(t_1)dt_1=1/((n-1)!)int_0^x(x-t)^(n-1)f(t)dt
(11)
 partial/(partialx_k)(x_jJ_k)=delta_(jk)J_k+x_jpartial/(partialx_k)J_k=J+rdel ·J
(12)
int_VJd^3r = int_Vpartial/(partialx_k)(x_iJ_k)-int_Vrdel ·Jd^3r
(13)
= -int_Vrdel ·Jd^3r
(14)

and the amusing integral identity

 int_(-infty)^inftyF(f(x))dx=int_(-infty)^inftyF(x)dx,
(15)

where F is any function and

 f(x)=x-sum_(n=0)^infty(a_n)/(x+b_n)
(16)

as long as a_n>=0 and b_n is real (Glasser 1983).

Integrals with rational exponents can often be solved by making the substitution u=x^(1/n), where n is the least common multiple of the denominator of the exponents.

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