Indefinite Integral of cos(x) / (1 - sin(x)): A Comprehensive Guide

This article provides a detailed explanation of finding the indefinite integral of the function cos(x) / (1 - sin(x)). This is a classic example of a rational trigonometric function, and understanding it thoroughly is crucial for calculus students and anyone dealing with integral calculus.

Introduction

One of the fundamental operations in integral calculus is finding the indefinite integral or anti-derivative of a given function. In this article, we will walk through the process of integrating the function cos(x) / (1 - sin(x)). This involves recognizing the form of the function and applying appropriate integration techniques.

Recognizing the Function

The function we are dealing with is cos(x) / (1 - sin(x)). This can be rewritten as:

[f(x) frac{cos(x)}{1 - sin(x)} frac{1 - sin(x)}{1 - sin(x)} cdot frac{cos(x)}{1 - sin(x)} frac{1 - sin(x)}{(1 - sin(x))^2} cdot cos(x) frac{1}{1 - sin(x)} cdot cos(x)]

This rearrangement helps us see the function clearly, and in the next step, we will use it to find the indefinite integral.

Using the Composite Compound Function

The derivative of a composite compound function of the form f(x) ln(u(x)) is given by:

[frac{d[ln(u(x))]}{dx} frac{u'(x)}{u(x)}]

Using this fact, we can see that the function we are integrating, cos(x) / (1 - sin(x)), can be viewed as a derivative of the natural logarithm of a certain function. Specifically:

[frac{d}{dx} [ln(1 - sin(x))] frac{-cos(x)}{1 - sin(x)}]

The negative sign is due to the chain rule, and we need to take the reciprocal and remove the negative sign:

[int frac{cos(x)}{1 - sin(x)} dx -int frac{-cos(x)}{1 - sin(x)} dx -ln(1 - sin(x)) C]

This means that the indefinite integral of the given function is:

[int frac{cos(x)}{1 - sin(x)} dx ln(1 - sin(x)) C]

Verification Using Substitution

To verify the result, consider the substitution method. Let:

[u 1 - sin(x)]

Then the differential is:

[du -cos(x) dx Rightarrow cos(x) dx -du]

Substituting these into the integral, we get:

[int frac{cos(x)}{1 - sin(x)} dx int frac{-du}{u} -ln|u| C -ln|1 - sin(x)| C]

Thus, the integral is equivalent to:

[int frac{cos(x)}{1 - sin(x)} dx ln(1 - sin(x)) C]

Conclusion

Through the application of substitution and the chain rule, we have found the indefinite integral of the function cos(x) / (1 - sin(x)) to be ln(1 - sin(x)) C. This process demonstrates the importance of recognizing the form of a function and applying appropriate integration techniques.

Key Takeaways

Recognize the form of the rational trigonometric function and apply the chain rule.

Use substitution to simplify the integral.

Verify the result by taking the derivative of the antiderivative.