Integration of 0 to π x / (1 - sin?x) dx: A Detailed Analysis

The integral in question is a fascinating problem in the realm of mathematical analysis. Specifically, we are dealing with the definite integral of the form:

$$I int_0^pi frac{x}{1 - sin x} , dx$$

Step-by-Step Solution

To solve this integral, we will employ a series of substitutions and algebraic manipulations. Let's begin by applying a trigonometric substitution and then utilize the property of symmetry in the interval.

First Approach: Symmetry and Interval Inversion

Given the integral: $$I int_0^pi frac{x}{1 - sin x} , dx$$ Apply the substitution: $$x mapsto pi - x$$, which leads to: $$I int_pi^0 frac{pi - x}{1 - sin(pi - x)} cdot (-dx) int_0^pi frac{pi - x}{1 - sin x} , dx$$. Average the two expressions of $$I$$: $$I frac{1}{2} int_0^pi left( frac{x}{1 - sin x} frac{pi - x}{1 - sin x} right) , dx frac{pi}{2} int_0^pi frac{1}{1 - sin x} , dx$$.

Second Approach: Symmetry and PYTHAGOREAN IDENTITY

Since the function $$f(x) sin x$$ is symmetric about the line $$x frac{pi}{2}$$ on the interval $$[0, pi]$$, we can simplify the bounds of integration:

Thus, we have: $$I frac{pi}{2} cdot 2 int_0^{pi/2} frac{1}{1 - sin x} , dx$$. Multiply numerator and denominator by $$1 - sin x$$ to use the Pythagorean identity: $$I pi int_0^{pi/2} frac{1 - sin x}{1 - sin^2 x} , dx pi int_0^{pi/2} frac{1 - sin x}{cos^2 x} , dx$$. Now, split the integral and use the substitution: $$int_0^{pi/2} sec^2 x - sec x tan x , dx pi cdot lim_{t to frac{pi}{2}^-} [tan x - sec x]_0^t$$. Evaluate the limit: $$I pi cdot lim_{t to frac{pi}{2}^-} (1 - tan t - sec t)$$.

Final Evaluation

Expressing the trigonometric functions in terms of sine and cosine, we get:

Thus, $$I pi cdot lim_{t to frac{pi}{2}^-} frac{sin t - 1}{cos t} pi cdot lim_{t to frac{pi}{2}^-} frac{cos t}{- sin t} pi$$.

Conclusion

The integral evaluates to $$I pi$$. This methodical breakdown showcases the elegance of trigonometric integrals and the power of symmetry in simplifying complex expressions.

References

For further study, you may refer to textbooks and online resources on integral calculus and trigonometric identities.