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Solved: Fractional powers Use the indicated substitution
Chapter 4, Problem 62E(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.
\(\int \frac{d x}{x \sqrt{1+2 x}}\) ; \(1+2 x=u^{2}\)
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.
\(\int \frac{d x}{x \sqrt{1+2 x}}\) ; \(1+2 x=u^{2}\)
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;