Trigonometric integrations play a pivotal role when integrating functions with squared trigonometric terms. This section sheds light on the techniques for integrating such functions, focusing on the use of trigonometric identities like and .
Utilising Trigonometric Identities
The integration of trigonometric functions often necessitates the application of identities to simplify the expressions. Double-angle formulas are particularly useful for converting squared trigonometric functions into forms more amenable to integration.
Double-Angle Formulas
The relevant double-angle identities are:
These can be utilised to simplify integrals of and .
Practical Example:
How do you calculate the area under the curve from to ?
Solution:
1. Identify the limits of integration:
- The function intersects the x-axis at and .
2. Implement the substitution :
- The differential is , implying .
3. Simplify the expression and integrate:
- The integral simplifies to , which reduces to .
4. Calculate the integral:
- The result of the integration is .
5. Apply the definite limits in terms of :
- At , , and at , .
- Evaluating the definite integral from to , we get , since the positive and negative areas cancel each other out.
Therefore, the area under the curve , denoted by , is confirmed to be .