bjohnson2001 said:I don't think it goes anywhere though, I get:
[itex]8cos(t)-4sin(t)cos(t)[/itex]
when it comes time to bring the original variable back in I get a mess of cos(arcsin(y/2)) but maybe I am missing something
Trig substitution is typically used when the integrand contains expressions involving the sine, cosine, or tangent functions. This could include expressions such as √(a^2-x^2), √(a^2+x^2), or √(x^2-a^2).
The basic steps for trig substitution are: 1) Identify which trigonometric function to substitute (based on the form of the integrand), 2) Express all other trigonometric functions in terms of the chosen one, 3) Substitute the trigonometric expression and its derivative into the integral, and 4) Simplify and solve the resulting integral.
Yes, there are a few special cases to keep in mind. For example, when the integrand contains √(a^2+x^2), we use the substitution x = a tanθ; when it contains √(x^2-a^2), we use the substitution x = a secθ; and when it contains √(a^2-x^2), we use the substitution x = a sinθ.
You may need to use trigonometric identities to simplify the integral after substitution. Be sure to review common identities such as sin^2θ + cos^2θ = 1 and sec^2θ = 1 + tan^2θ to help with simplification.
No, trig substitution is only applicable for certain types of integrals, particularly those involving trigonometric expressions. Other integration techniques, such as u-substitution or integration by parts, may be more appropriate for different types of integrals.