Solving the Integral of cos2x/1 sinx dx: A Step-by-Step Guide

In calculus, solving integrals can be a complex process, especially when dealing with trigonometric functions. One such challenge is the integral of cos2x/1 sinx dx. In this article, we will guide you through the steps to solve this integral, using detailed explanations and examples.

Understanding the Integral

The integral we want to solve is ∫ cos2x/1 sinx dx. At first glance, this can look daunting, especially for those who aren't familiar with trigonometric identities and integrals. However, with the right approach, we can simplify and solve it step by step.

Breaking Down the Problem

Let's start by rewriting the integral:

I ∫ cos2x/1 sinx dx

Using a trigonometric identity, we know that cos2x 1 - 2sin^2x. Therefore, we can rewrite our integral as:

I ∫ 1 - 2sin^2x/1 sinx dx

This can be further simplified using algebraic manipulation:

I ∫ 1 - sinx dx

Integration Process

Now, we can integrate each part separately:

I ∫dx - ∫sinx dx

This gives us:

I x - (-cosx) C

Simplifying, we get:

I x cosx C

Conclusion

Therefore, the integral of cos2x/1 sinx dx is x cosx C. This solution demonstrates the importance of using trigonometric identities and algebraic manipulation when solving integrals involving trigonometric functions.

Additional Tips

For those who are new to integral calculus, it is recommended to practice with similar problems and review basic trigonometric identities. This will make solving future integrals easier and more intuitive.

Keywords: integral cos2x, integration, trigonometric functions