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Differential Calculus finds Function .2/ from Function .1/. We recover the speedometer information from knowing the trip distance at all times. Integral Calculus goes the other way. The “integral” adds up small pieces, to get the total distance traveled. That integration brings back Function .1/.
- 7 TECHNIQUES OF INTEGRATION - MIT Mathematics
Note: A mnemonic device which is helpful for selecting when...
- 7 TECHNIQUES OF INTEGRATION - MIT Mathematics
A rational function is a fraction with polynomials in the numerator and denominator. For example, x3 1 x2 + 1. , , , x2 + x − 6 (x − 3)2 x2 − 1. are all rational functions of x. There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function.
Note: A mnemonic device which is helpful for selecting when using integration by parts is the LIATE principle of precedence for : Logarithmic Inverse trigonometric
Integrals. This chapter is about the idea of integration, and also about the technique of integration. We explain how it is done in principle, and then how it is done in practice. Integration is a problem of adding up infinitely many things, each of which is infinitesimally small. Doing the addition is not recommended.
Techniques of Integration. Chapter 5 introduced the integral as a limit of sums. The calculation of areas was started—by hand or computer. Chapter 6 opened a different door. Its new functions exand lnxled to differential equations.
A powerful class of techniques is based on the observation made at the end of chapter 6, where we saw that the fundamental theorem of calculus gives us a second way to find an integral, using antiderivatives.
Contents Preface xvii 1 Areas, volumes and simple sums 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Areas of simple shapes ...
