The integral of cos(ax) is (1/a)sin(ax) + C, where C is the constant of integration.
Here's how we arrive at this solution:
- Understanding the Derivative: The derivative of sin(ax) is a*cos(ax). This relationship is key to finding the integral of cos(ax).
- Reversing the Derivative: To find the integral, we essentially reverse the process of differentiation. Since the derivative of sin(ax) includes the factor 'a', we need to divide by 'a' when integrating cos(ax).
- Constant of Integration: Remember that the derivative of a constant is always zero. This means that any constant 'C' added to the integral (1/a)sin(ax) would still have a derivative of cos(ax). Hence, we include '+ C' to represent any possible constant term.
Example:
Let's say we want to find the integral of cos(2x):
- Using the formula: The integral of cos(2x) is (1/2)sin(2x) + C.
Practical Insights:
- This integral is fundamental in calculus and has applications in various fields, including physics, engineering, and signal processing.
- The constant of integration 'C' is important because it allows us to represent a family of functions that all have the same derivative.
