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.

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

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.