Fractional powers Use the indicated substitution to

Chapter 4, Problem 63E

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QUESTION:

Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral.

\(\int \frac{d x}{\sqrt{x}+\sqrt[3]{x}}\) ; \(x=u^{6}\)

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QUESTION:

Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral.

\(\int \frac{d x}{\sqrt{x}+\sqrt[3]{x}}\) ; \(x=u^{6}\)

ANSWER:

Step 1 of 5

             Definition of a Rational Function;  A rational  function is a function that is a fraction  and has a property that  both its  numerator  and denominator  are polynomials . In other words , R(x) is  a rational function if R(x) =  where p(x) and q(x) are both polynomials , and q(x) recall that a polynomial is any function of the form  f(x) = a +bx+ c+.............. +n, where a,b , c ……………….n  are all real numbers and the exponents of each x is a non -negative integer.

            Proper fraction definition ; In a rational fraction  , if the degree of f(x) < the degree of g(x) , then the rational fraction is called a proper fraction.

         The sum of two proper fractions is a proper fraction.

                                  Example;

             Improper fraction definition; In a rational fraction  , if the degree of f(x)  the degree of g(x) , then the rational fraction is called an improper fraction.

           If an improper rational fraction  is given for splitting into partial fractions , we first divide f(x) with g(x) till we obtain a remainder R(x) of lower degree than g(x).

          First we express the fraction in the form  = quotient +

         Then we resolve the final proper fraction into partial fractions.

           

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