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Integration Techniques In our journey through integral calculus, we have: developed the con-cept of a Riemann sum that converges to a definite integral; learned how to use the Fundamental Theorem of Calculus to evaluate a definite integral—as long as we can find an antiderivative for the integrand;
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/.
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.
With integration by parts, and a new substitution, they become simple. Those examples indicate where this chapter starts and stops. With reasonable effort (and the help of tables, which is fair) you can integrate important functions. With intense effort you could integrate even more functions.
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.
A few molecules like water or methane should be known without √ having to look it up. Functions to know to integrate are polynomials like x5 or x, rational functions like 1/x or 1/(1 + x2 or trig functions like sin, cos, tan and the ex-ponential exp and the logarithm log.
