Integration with Trigonometric Substitution is a method of solving integrals that involve trigonometric functions by using a substitution technique. It is particularly useful for integrals containing functions like sine, cosine, and tangent.
Trigonometric Substitution is most useful when the integrand contains a combination of algebraic and trigonometric functions, such as √(x²+1) or √(1-x²). It can also be used when the integrand contains expressions with radicals, such as √(a²-x²) or √(x²-a²).
The general steps for using Trigonometric Substitution are:
1. Identify the type of substitution needed based on the form of the integrand.
2. Substitute the given variable with the corresponding trigonometric function.
3. Use trigonometric identities to simplify the integral.
4. Solve the new integral using standard integration techniques.
5. Substitute back the original variable to obtain the final answer.
The most commonly used trigonometric substitutions are:
1. For integrals with expressions containing √(a²-x²), use x = a sinθ or x = a cosθ substitution.
2. For integrals with expressions containing √(x²+a²), use x = a tanθ or x = a secθ substitution.
3. For integrals with expressions containing √(x²-a²), use x = a secθ or x = a tanθ substitution.
No, Trigonometric Substitution is not always applicable to solve integrals. It is only useful for certain types of integrals that involve trigonometric functions. For other integrals, different techniques such as integration by parts or partial fractions may need to be used.
