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We've covered the most important rules and methods for integration already. We'll look at a few special-purpose methods later on. The fundamental theorem of calculus. This is the most important theorem for integration. It tells you that in order to evaluate an integral, look for an antiderivative.
Integrals of Exponential and Logarithmic Functions. ∫ ln x dx = x ln x − x + C. + 1 x. + 1. x ∫ x ln xdx = ln x − + C. 2 + 1 ( n + 1 ) x dx = e x + C ∫.
There are two major ways to manipulate integrals (with the hope of making them easier). Substitutions are based on the chain rule, and more are ahead. Here we present the other method, based on the product rule. The reverse of the product rule, to find integrals not derivatives, is integration by parts.
Evaluating integrals by applying this basic definition tends to take a long time if a high level of accuracy is desired. If one is going to evaluate integrals at all frequently, it is thus important to find techniques of integration for doing this efficiently.
basic rules of integration, as well as several common results, are presented in the back of the log tables on pages 41 and 42. Rule 1 Since the derivative of xn+1 is (n+1)xn it follows that Z xndx = xn+1 n+1 +c provided n 6= 1. Thus to integrate a power of x, we increase the power by 1 and divide by the new power. This rule can be used to ...
There are certain methods of integration which are essential to be able to use the Tables effectively. These are: substitution, integration by parts and partial fractions. In this chapter we will survey these methods as well as some of the ideas which lead to the tables.
Integration Rules and Formulas Properties of the Integral: (1) Z b a f(x)dx = Z a b f(x)dx (2) Z a a f(x)dx = 0 (3) Z b a kf(x)dx = k Z b a f(x)dx (4) Z b a [f(x)+g(x)]dx = Z b a f(x)dx+ Z b a g(x)dx (5) Z b a f(x)dx = Z c a f(x)dx+ Z b c f(x)dx (a < c < b) (6) Z b a F0(x)dx = F(b) F(a) (7) d dx Z x a f(t)dt = f(x) (8) d dx Z g(x) a f(t)dt = f ...
