Subsections

# Rational Functions and Partial Fractions

## Purpose

The purpose of this lab is to show how quotient functions can be integrated.

## Rational Functions

When a function is the quotient of two polynomials, you can easily take the integral if the numerator is the derivative of the denominator.

> diff(x^2+2*x+9,x);

Note that the numerator times a constant is the derivative of the denominator.
> simplify((6*x+6)/(2*x+2));

> int(3/u,u);
> subs(u=x^2+2*x+9,int(3/u,u));

To check the work, let Maple do the intgral directly.
> int((6*x+6)/(x^2+2*x+9),x);

Remember with indefinite integrals the solution adds a constant. So, the inetgral solution is . Often a function is not in that straight forward form. With long division, you can try and get the quotient function into the form of a polynomial plus a fraction where the numerator is a derivative of the denominator: . For example if

First execute long division and find the quotient and remainder.
> q:=quo((x^3+x^2+x-1),(x^2+2*x+2),x);
> r:=rem((x^3+x^2+x-1),(x^2+2*x+2),x);

The new form of the function is:
> f:=q+r/(x^2+2*x+2);

Note that the fractional part has the numerator a derivative times a constant of the denominator.
> diff(x^2+2*x+2,x);
> simplify(r/diff(x^2+2*x+2,x));
> int(q,x)+subs(u=x^2+2*x+2,int(1/(2*u),u));

To check the work, let Maple do the integral directly.
> int((x^3+x^2+x-1)/(x^2+2*x+2),x);


## Partial Fractions

When the function is a fraction with a denominator that can be factored into linear components then the partial method can be easily used.

The denominator is easily factored:
> factor(x^2-1);

So, . Multiplying by the common denominator and expanding gives:
> expand(5*x-1=A*(x-1)+B*(x+1));

With this equation we can solve for and by equating the coefficients of the x term and then equating the constants. This will give us two equations which can be solved simultaneously.
> solve({5*x=A*x+x*B,-1=-A+B},{A,B});

These values tell us that: . The right-hand side shows fractions that are easily integrated with the natural log.
> int(3/(x+1)+2/(x-2),x);

To check the work let Maple do the integral directly.
int((5*x-1)/(x^2-1),x);

Remember the constant: .

## Exercises

Evaluate each of the following definite integrals below using long division and/or partial fractions. Show all steps and include plenty of text to keep your work clear. Also check your final answer by having Maple do the integral directly.