Integrating the Expression ( frac{dx}{1 - sin x} ): A Comprehensive Guide

In this article, we will provide a detailed step-by-step guide to integrate the expression ( frac{dx}{1 - sin x} ). We will explore multiple methods including the use of trigonometric identities and substitutions. Let's dive into the process.

Method 1: Conjugate Multiplication and Trigonometric Substitution

The first approach involves multiplying the numerator and denominator by the conjugate of the denominator, which simplifies the integrand for easier integration.

Step 1: Multiply by the Conjugate

To simplify the integrand, multiply the numerator and the denominator by the conjugate of the denominator:

(int frac{dx}{1 - sin x} cdot frac{1 sin x}{1 sin x} int frac{1 sin x}{1 - sin^2 x} , dx int frac{1 sin x}{cos^2 x} , dx)

Step 2: Split the Integral

Next, split the integral into two parts:

(int frac{1 sin x}{cos^2 x} dx int frac{1}{cos^2 x} dx int frac{sin x}{cos^2 x} dx)

Step 3: Solve Each Integral

For the first integral:

(int frac{1}{cos^2 x} dx tan x C_1)

For the second integral, use the substitution (u cos x), which gives (du -sin x , dx) or (-du sin x , dx):

(int frac{sin x}{cos^2 x} dx -int frac{1}{u^2} du -left(frac{-1}{u}right) C_2 frac{1}{u} C_2 sec x C_2)

Step 4: Combine Results

Combining both results, we get:

(int frac{dx}{1 - sin x} tan x sec x C)

Method 2: Trigonometric Substitution

Alternatively, we can substitute (1 - sin x) with a different form to make the integral more straightforward.

Step 1: Express 1 - sin x in Terms of cos(pi/2 - x)

First, express (1 - sin x) in terms of (cos(pi/2 - x)):

(1 - sin x 1 - cos(pi/2 - x))

Step 2: Substitute t (pi/2 - x)

Let (t pi/2 - x), then (dt -dx):

(int frac{dx}{1 - sin x} int frac{-dt}{cos t})

Since (cos t cos(pi/2 - x) sin x), we have:

(int frac{dx}{1 - sin x} -int frac{dt}{sin x})

Step 3: Use the Substitution (t/2 u)

Using the identity (cos t 1 - 2sin^2(u)), let (t/2 u), then (dt 4du):

(int frac{1 - 2sin^2(u)}{1 - (1 - 2sin^2(u))} cdot 4du 4int frac{1}{cos^2 u} du 4tan u C 4tan(pi/4 - x/2) C)

Final Answer

Thus, the integral of (frac{dx}{1 - sin x}) is:

(int frac{dx}{1 - sin x} tan(x) sec(x) C)

Conclusion

In this comprehensive guide, we have discussed two methods to integrate the expression (frac{dx}{1 - sin x}). Each method offers a unique perspective, making the integration process more manageable and understandable.