Proving Trigonometric Integrals through Substitution and Simplification

Trigonometric integrals, a category of integrals that involves trigonometric functions, can often be simplified and evaluated through strategic substitutions and identities. This article will illustrate how to prove the integral of a specific function using these techniques, providing a step-by-step guide that aligns with Google's SEO standards.

Introduction to the Trigonometric Identity

In this discussion, we will be dealing with the integral:

I int_0^{2pi} frac{dtheta}{abcostheta c sintheta}

Step 1: Simplifying the Integrand Using Trigonometric Identities

The first step is to use trigonometric identities to simplify the integrand. Specifically, we will use identities that relate sine, cosine, and tangent. The integral can be rewritten as follows:

I int_0^{2pi} frac{dtheta}{ab frac{1-tan^2theta}{1 tan^2theta} c frac{2tan theta}{1 tan^2theta}}

Simplifying the integrand further, we get:

I int_0^{2pi} frac{sec^2 theta dtheta}{ab - btan^2 theta 2ac tan theta - ab}

By reorganizing the terms, we can simplify the denominator:

I int_0^{2pi} frac{sec^2 theta dtheta}{a - btan^2 theta 2ac tan theta - ab}

Step 2: Substitution for Simplification

The next step is to perform the substitution (tan theta x). This gives:

I frac{1}{a - b} int_0^{2pi} frac{dx}{x left(frac{2c}{a-b}x frac{ab}{a-b}right)}

Further simplification is now straightforward. By completing the square in the denominator, we obtain:

I frac{1}{a - b} int_0^{2pi} frac{dx}{x left(frac{c}{a-b}right)^2 frac{a^2 - b^2 - c^2}{(a-b)^2}}

This setup allows us to use the inverse tangent function as follows:

I boxed{frac{1}{a - b} cdot frac{1}{k} tan^{-1} left(frac{tan theta cdot frac{c}{a-b}}{k}right)}

where (k frac{sqrt{a^2 - b^2 - c^2}}{a-b}).

Step 3: Dividing the Integral Over Subintervals

To evaluate the integral correctly, it is necessary to divide the interval ([0, 2pi]) into subintervals to maintain the continuity of the integrand. This is because the tangent function is not continuous at (theta frac{pi}{2}) and (theta frac{3pi}{2}). Therefore, we divide the integral into four parts:

I frac{1}{a - b} int_0^{frac{pi}{2}} frac{dx}{x left(frac{c}{a-b}right)^2 frac{a^2 - b^2 - c^2}{(a-b)^2}}

frac{1}{a - b} int_{frac{pi}{2}}^{pi} frac{dx}{x left(frac{c}{a-b}right)^2 frac{a^2 - b^2 - c^2}{(a-b)^2}}

frac{1}{a - b} int_{pi}^{frac{3pi}{2}} frac{dx}{x left(frac{c}{a-b}right)^2 frac{a^2 - b^2 - c^2}{(a-b)^2}}

frac{1}{a - b} int_{frac{3pi}{2}}^{2pi} frac{dx}{x left(frac{c}{a-b}right)^2 frac{a^2 - b^2 - c^2}{(a-b)^2}}

Evaluating these integrals separately, we get:

I boxed{frac{2pi}{sqrt{a^2 - b^2 - c^2}}}

Conclusion

The process of proving the integral of a trigonometric function involves a combination of simplifying the integrand using trigonometric identities, performing a strategic substitution, and dividing the integral into subintervals to maintain the continuity of the integrand. This method provides a clear and comprehensive framework for evaluating integrals of this type.