What is the process for integrating sin(2x)?

by

Understanding the Integration of sin(2x)

Integrating the function sin(2x) involves applying basic integration techniques from calculus. The goal is to find the antiderivative or integral of the function with respect to x.

Steps to Integrate sin(2x)

  1. Identify the Integral: You want to compute ∫ sin(2x) \, dx.
  2. Use Substitution: To simplify the process, we can use the substitution method. Let u = 2x. This means that du = 2 \, dx, or dx = \frac{du}{2}.
  3. Change the Variable: Substitute back into the integral: ∫ sin(2x) \, dx = ∫ sin(u) \, \frac{du}{2}. This simplifies to \frac{1}{2} ∫ sin(u) \, du.
  4. Integrate sin(u): The integral of sin(u) is -cos(u), so you have: \frac{1}{2} (-cos(u)) + C, where C is the constant of integration.
  5. Back Substitute: Replace u back with 2x: -\frac{1}{2} cos(2x) + C.

Final Result

The integral of sin(2x) with respect to x is:

∫ sin(2x) \, dx = -\frac{1}{2} cos(2x) + C

This result can be useful in various applications, such as solving differential equations or analyzing oscillatory motion.

Conclusion

Understanding how to integrate functions like sin(2x) deepens your knowledge of calculus and its applications. Always remember to check your work by differentiating your result to ensure you get back the original function.

Leave a Comment