Improper integrals

There are certain restrictions of the integration definition, The func-tion/(s) must be continuous in the finite intei val ia, b) with at most a finite number of finite discontinuities, which must be obsei ved before integration formulas can be generally apphed. Two of these restrictions give rise to so-called improper integrals and require special handling. These occur when... 

If one (or both) of the limits of integration is infinite, or if the integrand itself becomes infinite at or between the limits of integration, the integral is an improper integral. Depending on the function, the integral may be defined, may be equal to < , or may be undefined for all x or for certain values of x. 

Improper integrals of the other types whose problems involve both limits are handled by open formulas that do not require the integrand to be evaluated at its endpoints. One such formula, the extended midpoint rule, is accurate to the same order as the extended trapezoidal rule and is used when the limits of integration are located halfway between tabulated abscissas ... 

Z) is a triangle. The integral in (4,75) is, in Riemann s definition, an improper integral. It can be approximated by a finite sum because the integrand is monotone, which implies, furthermore, that is bounded. I skip the details because considerations of this type are standard in the discussion of improper Riemannian integrals, Equations (4,73) and (4.75), in the notation of (4.56), imply... 

Maclaurin s Integral Test. Suppose Is, is a series of positive terms and/is a continuous decreasing function such that/(.r ) > 0 for 1 < x < so and/(n) = an. Then the series and the improper integral j fix) dx either both converge or both diverge. 

The Laplace transform of a function//) is defined by F(s) = L f(t) = I(Te s/(f) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions/for which this improper integral converges. The Laplace transform is used in process control (see Sec. 8). 

Figure A.7 Demonstration of integration for a domain with two poles on the real axis. A.3.2 Improper Integrals of Rational Functions...

Trapezoidal rule Simplicity. Optimal for improper integrals. Needs a large number of subintervals for good accuracy. 

There are two types of improper integrals. One type of improper integral is that where the integrand becomes infinite at one of the limits of integration. The other type is that where one or both of the limits of integration are infinite. The improper integral is defined as... 

If the limit exists, then the improper integral converges otherwise, the improper integral diverges. Given/(x) on [a,b] such that/(a) = then the improper integral is defined as... 

The improper integrals are said to be convergent if and only if the limits are convergent. 

The improper integral can be calculated by passing from the variable 77 to the... 

An improper integral has at least one infinite limit or has an integrand function that is infinite somewhere in the interval of integration. If an improper integral has a finite value it is said to converge. Otherwise it is said to diverge. 

So far we have assumed that both limits of a definite integral are finite and that the integrand function does not become infinite inside the interval of integration. If either of these conditions is not met, an integral is said to be an improper integral. For example. 

Another kind of improper integral has an integrand function that becomes infinite somewhere in the interval of integration. For example. 

The two principal questions that we need to ask about an improper integral are ... 

EXAMPLE 5.9 Determine whether the following improper integral converges, and if so, find its value ... 

Determine whether the following improper integrals converge. Evaluate the convergent integrals. 

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