Final answer:
To express the integrand as a rational function, make a substitution and simplify the expression. Then, evaluate the integral using appropriate integration techniques.
Step-by-step explanation:
To express the integrand as a rational function, we can make a substitution. Let's say we have the integral ∫ f(x) dx and we want to convert it into a rational function of u. We can substitute x with a function u(x) and dx with the derivative du/dx, which gives us ∫ f(u(x)) du/dx du. By multiplying and dividing by du/dx, we can simplify the expression as ∫ f(u) du, where f(u) is a rational function of u.
Once we have expressed the integrand as a rational function, we can evaluate the integral by using techniques such as partial fractions, trigonometric substitutions, or integration by parts. The specific method will depend on the form of the rational function.
For example, if we have the integral of (x^2 + x + 1) / (x + 1), we can divide the numerator by the denominator to obtain x + 1 + 1/(x + 1). We can then integrate each term separately to evaluate the integral.