Trig substitution is a method used to simplify integrals that involve square roots of quadratic expressions. It involves substituting a trigonometric function for a variable in the integral, which allows for easier integration.
Trig substitution is typically used when the integrand (the expression being integrated) contains a square root of a quadratic expression, or when the integrand contains a sum or difference of squares.
The appropriate trig substitution can be determined by looking at the form of the integrand. For a square root of a quadratic expression, use a substitution involving secant or tangent. For a sum or difference of squares, use a substitution involving sine or cosine.
No, trig substitution is not always applicable to all integrals. It is most useful for integrals that involve square roots of quadratic expressions or sums and differences of squares. Other methods, such as integration by parts or partial fractions, may be more appropriate for other types of integrals.
One common mistake is forgetting to substitute back the original variable after integrating. It is also important to carefully choose the appropriate trig substitution and to use the appropriate trig identities to simplify the integral.