The integral of cos 2 theta is (1/2)sin 2 theta + C, where C is the constant of integration.
Here's a breakdown of how we arrive at this answer:
- The integral of cos x is sin x + C. This is a fundamental trigonometric integral you should be familiar with.
- We need to account for the coefficient of 2 in our integrand (cos 2 theta). To do this, we use the chain rule in reverse. Essentially, we divide by the derivative of the inside function (2 theta), which is 2.
- Therefore, the integral of cos 2 theta is (1/2)sin 2 theta + C.
Let's illustrate this with an example:
Example:
Find the integral of cos 2x with respect to x.
- Step 1: We know the integral of cos x is sin x + C.
- Step 2: We need to account for the coefficient of 2 in our integrand (cos 2x). We divide by the derivative of the inside function (2x), which is 2.
- Step 3: Therefore, the integral of cos 2x is (1/2)sin 2x + C.
Practical Insights:
- Understanding this integral is crucial in various fields like physics, engineering, and mathematics.
- It often arises when dealing with problems involving oscillations, waves, and periodic phenomena.
