Problem 62E

Fractional powers

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

Problem 62E

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

Answer;

Step 1

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;

Partial fractions Depending upon the nature of factors of Denominator ;

- When the denominator has non-repeated linear factors;

A non - repeated linear factor (x-a) of denominator corresponds a partial fraction of the form , where A is a constant to be determined’

If g(x) = (x-a)(x-b)(x-c)............(x-n), then we assume that

= ++ +...............+

Where A, B, C,............N are constants which can be determined by equating the numerator of L.H.Sto the numerator of R.H.S , and substituting x = a,b ,c ….n.