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 $sin²(x)$ and $cos²(2x)$.

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 $sin²(x)$ and $cos²(x)$.

Practical Example:

How do you calculate the area under the curve $y = \sin^3(2x)\cos^3(2x)$from $x = 0$ to $x = \frac{\pi}{2}$?

Solution:

1. Identify the limits of integration:

• The function intersects the x-axis at $x = 0$ and $x = \frac{\pi}{2}$.

2. Implement the substitution $u = sin(2x)$:

• The differential is $du = 2cos(2x)dx$, implying $dx = \frac{du}{2cos(2x)}$.

3. Simplify the expression and integrate:

• The integral simplifies to $\int sin³(2x)cos²(2x) \cdot \frac{du}{2cos(2x)}$, which reduces to $\frac{1}{2} \int u³(1 - u²) du$.

4. Calculate the integral:

• The result of the integration is $-\frac{u^6}{12} + \frac{u^4}{8}$.

5. Apply the definite limits in terms of $u$:

• At $x = 0$, $u = 0$, and at $x = \frac{\pi}{2}$, $u = 1$.
• Evaluating the definite integral from $u = 0$ to $u = 1$, we get $0$, since the positive and negative areas cancel each other out.

Therefore, the area under the curve $y = sin³(2x)cos³(2x)$, denoted by $A$, is confirmed to be $0$.