# Complex integration of real-valued trig function

• brandones
In summary, the conversation is about finding the integral of 1/(3+2cos(θ)) from zero to pi. The speaker suggests using complex tricks and variable changes to solve the integral, but if those are not possible, they suggest calculating the indefinite integral and plugging in the values of theta.
brandones

## Homework Statement

Integrate:

$$\int \frac{1}{(3+2cos(θ))} dθ$$ evaluated from zero to pi.

## Homework Equations

I can't think of any. All of the integration formulas in the text rely on the existence of a singularity somewhere in the complex plane. This thing is analytic everywhere.

## The Attempt at a Solution

Since it's analytic everywhere, its integral over any closed loop equals zero. That tells me that its integral along the x-axis from zero to pi is equal to the negative of its integral along any other path from pi to zero. I tried to find a path along which cos(theta) is linear (which would give it average value zero) to no avail. I have no idea where to go.

Your integral gives Pi/sqrt(5), calculate the plain ol' indefinite integral and plug in theta=Pi and theta=0.

If you have to do it using complex tricks, I suggest a variable change t = theta-pi/2, which gives $$\int_{-\pi/2}^{\pi/2} \frac{1}{3 - 2 \sin t} \mathrm{d} t$$, and then you can stretch your integration interval using u = tan t. This gives you an integral from -infty to +infty which has singularities for I am u \neq 0. I'm sure you know how to do this then, if not, just ask again.

## What is a complex integration?

A complex integration is a mathematical process that involves finding the area under a curve in the complex plane. It is similar to finding the area under a curve in the real plane, but it takes into account the imaginary component of the function.

## What is a real-valued trig function?

A real-valued trig function is a trigonometric function that outputs real numbers. Examples of real-valued trig functions include sine, cosine, and tangent.

## Why is complex integration of real-valued trig functions important?

Complex integration of real-valued trig functions is important because it allows us to solve problems involving complex numbers and trigonometric functions. It is also a fundamental tool in various branches of mathematics and physics.

## What are some techniques for solving complex integration of real-valued trig functions?

Some techniques for solving complex integration of real-valued trig functions include using substitution, integration by parts, and trigonometric identities. It is also helpful to have a good understanding of complex numbers and their properties.

## How is complex integration of real-valued trig functions used in real-world applications?

Complex integration of real-valued trig functions is used in various fields, such as engineering, physics, and economics. It is used to model and solve problems involving alternating currents, vibrations, and oscillations. It is also used in signal processing and image processing.

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