Integrals - Integrals From Physics Notes Here is a brief...

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Integrals From Physics Notes Here is a brief summary of the integral calculus. Contents 1 Definite integrals and antiderivatives 2 Integration by parts 3 Change of variables 4 The Gaussian integral 5 Differentiating under the integral sign 6 Exercises Definite integrals and antiderivatives If we have a function which is well-defined for some , its integral over those two values is defined as This is called a definite integral , and it represents the area under the graph of in the region between and , as shown in Fig. 1. For the purposes of dimensional analysis, an integral has the units of the integrand times the units of (this is easy to remember: just treat as a multiplicative factor with units of ). From the defintion of the derivative, we can show that Hence, an integral is the "inverse" of a derivative operation. Notice that the right-hand-side of the first equation does not involve , the opposite integral limit. Based on this, we can define an indefinite integral , or antiderivative : Unlike a definite integral, an antiderivative is not unique, but is only defined up to an additive constant (called an integration constant ). As you may recall, integration is much harder than differentiation. Once you know how to differentiate a few special functions, differentiating some combination of those functions just involves a straightforward (though possibly tedious) application of composition rules. By contrast, there is no general systematic procedure for doing
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Fig. 1: A definite integral of a function can be defined as a sum of rectangle areas. The interval between two fixed points, and , is divided into segments of length . For an integral symbolically. This is called the "antiderivative problem" ( ) .
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