Search results
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.
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.
Note: A mnemonic device which is helpful for selecting when using integration by parts is the LIATE principle of precedence for : Logarithmic Inverse trigonometric
For integrals there are two steps to take—more functions and more applications. By using mathematics we make it live. The applications are most complete when we know the integral. This short chapter will widen (very much) the range of functions we can integrate. A computer with symbolic algebra widens it more.
1.1.2. Evaluating Integrals. We will soon study simple and ef-ficient methods to evaluate integrals, but here we will look at how to evaluate integrals directly from the definition. Example: Find the value of the definite integral R1 0 x2 dx from its definition in terms of Riemann sums.
Integration by parts. 4.2. Write the integrand as a product of two functions, diferentiate one u and inte-grate the other dv. Then use R udv = uv − R vdu from the product formula. Example: cos(x/3) dx. Solution: diferentiate u = x and integrate dv = cos(x/3)dx. We have 3x sin(x/3) −. R 1 · 3 sin(x/3) = 3x sin(x/3) + cos(x/3)9 + C. Remarks.
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/.
