A New Approach to Evaluating the Integral of 1 over (1 cos θ cos x) from 0 to 2π: An Optimized Method for SEO

This article presents a detailed, optimized approach to evaluate the integral of the function I∫02π11 cosθcosxdx. The traditional method often involves complex transformations, whereas this new approach provides a clearer and more straightforward solution.

New Method

The integral of interest is given as:

∫02π11 cosθcosxdx

Let us present it in a step-by-step manner:

Clean and simplified form: ∫02π11 cosθcosxdx

A corrected expansion: ∫02π11 cosθcosxdx∫02π11-cosθcosxdx-cosθ∫cosx1-cosθcosxdx

Further simplification: ∫02π11-cosθcosxdx and -cosθ∫cosx1-cosθcosxdx, then integrating based on the given formulae.

Substitution: x→tanx2, simplifying the integral as:

Final result: I2πsinθ

The provided solution involves breaking down the integral into simpler components, leveraging substitution and integrating trigonometric functions to achieve the desired result.

Original Method

The original method involves dividing the interval and leveraging a substitution. The integral is redefined as:

I∫02π11 cosθcosxdx

Simplification through substitution ( y 2pi - x ): I∫0π11 cosθcosxdx ∫π2π11 cosθcosxdx

Through simplification and integration:

I2∫0π11 cosθcosxdx

Further simplification using ( t tan(frac{x}{2}) ): I2∫0∞11 cosθcosxdx2∫0∞11-cosθt2-1t2dx

Finally, evaluating the integral:

I2πsinθ

The original method also leverages substitution and simplification to reach the final result.

Conclusion

In both methods, the key steps involve simplification through substitution, leveraging properties of trigonometric functions, and evaluating the integral. The new method provides a clearer path, making it more accessible for implementation in various settings, particularly for SEO purposes in mathematical discussions.

Keywords

Meta Description

This article details a new approach to evaluate the integral of the function 1/(1 cos θ cos x) from 0 to 2π, using straightforward and systematic steps. This method is optimized for SEO and provides clear insights into integral evaluation techniques.

Tags: Integral, Trigonometric Integral, Mathematical Optimization