Question

Use in your own words to discuss how you solve the integration of rational functions. Make sure that you include all methods which we have discussed in our class.

Answer 1

ANSWER

In order to convert improper rational function into a proper one, we can use long division: P(x)Q(x)=F(x)+R(x)Q(x), where F(x) is a polynomial, R(x)Q(x) is a proper rational function. To integrate a proper rational function, we can apply the method of partial fractions.

* The denominators of the partial fractions can contain non repeated linear factors, repeated linear factors, non repeated irreducible quadratic factors and repeated irreducible Answelt Integration of Rational Function + let the rational function is a ratio of two polynomials P(x) → We will assume that we have a propere rational QC). function in which the degree of the numerator is less than the degree of the denominator. → In order to convert improper rational function into a proper one, we can use long division:- P (W) OCK) where Flr) is a polynomial, RCU) is a proper OCX) rational function. > Po integrate aproper rational function, we can apply the method of partial fractions → This method allows to turn the integral of a complicated rational function into the sun of integrals s of simplert functions. quadratic factors = F(X) + R (X) (X)

A y to evaluate integrals of partial fractions with linear or quadratic denom inators, We use the following o formulas: Adn Aln lantb] antb Adn Cantb] h 2(1-15) Cax +b)85-1 For partial fractions with irreducible quadratic denominators, we first complete the square: ax²+buto - a 4ac-b2 ut 4a2 we can write AntB dn At+B! {alt?+m2)]** adt Satt ats At + B! (+2+ m2), dt. where t=x + b m34ac-b2 2a 402 B'- B - Ab 29 The possible cates for fractions with quadratic de nominator are covered by the following Integrals:

3. tat Linctum) +2+ m2 dt S +2+m2 I arctan t m m 5. (+2+m2) k 1 20- )(+2+m-]K-1 Finally, the integral se tamoyi can be evaluated in k steps us’ing the reduction formula. at (+24m2 6. t + 2k-3 at 2m2(K-1) (+2+ m2)8-) 2m 945-1) (+44m2) K-1 Example: 2) find the Integral S**2 d. Sol) As the rational fraction in the in te grand is impropee we perform a long division to get 1+1 =1+ 3 -1 n-1 then Pinte easily evaluate the integrali Just don = / (1+2 Jan. = fontslae te **+31n1nailta Answer →|+31n 110-11 +c