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Q: calculus integrations ( Answered,   1 Comment )
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 Subject: calculus integrations Category: Science > Math Asked by: kdmac-ga List Price: \$10.00 Posted: 08 Jun 2006 17:27 PDT Expires: 08 Jul 2006 17:27 PDT Question ID: 736544
 ```Integral calculus. Solve with explanations 2 integrations. 1. "x^2+1/x^2" and "x^3-1/x-1". The method is the important part.```
 Subject: Re: calculus integrations Answered By: livioflores-ga on 08 Jun 2006 21:14 PDT
 ```Hi!! Since the commenter suggests, the best method to find the antiderivatives of "x^2+1/x^2" and "x^3-1/x-1" is by simplifying the rational expressions and then apply the basic rules of integration. 1-. "x^2+1/x^2" (x^2+1)/x^2 = 1 + 1/x^2, then: (int means integral) int(x^2+1)/x^2 dx = int (1 + 1/x^2) dx = = int dx + int 1/x^2 dx = = (x+C) + (-1/x+D) = = x - 1/x + E where C, D, E are unknown constants and E=C+D . 2-. "x^3-1/x-1" Note that x=1 is a root of (x^3-1), the (x-1) is a divisor of (x^3-1): x^3 - 1 = (x-1)*(x^2 + x + 1) Then: (x^3 - 1) / (x-1) = (x^2 + x + 1) , so you have that: int (x^3-1/x-1)dx = int (x^2 + x + 1)dx = = int x^2 dx + int x dx + int dx = = x^3/3 + x^2/2 + x + K where K is an unknown constant. The following pages will help you to better understand how to simplify rational expressions: "Simplify Rational Expressions": http://www.analyzemath.com/Rational_expressions/Simp_rat_expre.html "College Algebra Tutorial on Simplifying Rational Expressions": http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htm "Rational Functions": http://daphne.palomar.edu/mmumford/5660notesPDF/chap8.pdf Search strategy: My own knowledge when solving the problems adn then use the following keywords at Google.com: simplifying rational expressions I hope this helps you. feel free to request for a clarification if you need it. Regards, livioflores-ga```
 Subject: Re: calculus integrations From: kingal-ga on 08 Jun 2006 20:15 PDT
 ```1) I assume you mean (x^2 + 1)/x^2. Simplify to 1 + 1/x^2, then integrate. 2) Divide x^3 -1 by x-1, then integrate that.```