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Fractional powers Use the indicated substitution to
Chapter 4, Problem 63E(choose chapter or problem)
Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral.
\(\int \frac{d x}{\sqrt{x}+\sqrt[3]{x}}\) ; \(x=u^{6}\)
Questions & Answers
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.