Question

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{-\pi / 4}^{3 \pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} $$

Short answer

\(\frac{\pi}{\sqrt{2}}\)

Step-by-step solution

Identify the Integrand

The given integral is \[ \int_{-\pi / 4}^{3 \pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} \] Identify the function inside the square root in the denominator.

Simplify the Integrand Function

Recognize that \(1 + \cos^2(\phi)\) is a function that if simplified further isn't straightforward to integrate using elementary functions. However, observe the symmetry properties over the given limits.

Apply Symmetry

Notice that \(1 + \cos^2(\phi)\) is an even function because \(\cos^2(\phi) = \cos^2(-\phi)\). We can use the property of even functions over symmetric limits to simplify the integral range:

Change of Limits

Double the integral from 0 to \(\pi / 4\) instead of \([-\pi / 4, 3 \pi / 4]\) since the function is symmetric: \[ \int_{-\pi / 4}^{\pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} = 2 \int_0^{\pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} \]

Using Known Integral Forms

Identify that \(\int_0^{\pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}}\) does not have an elementary antiderivative but there are known results or tables that provide the integral value. Using elliptic integral reference tables, the value of: \[ \int_0^{\pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} = \frac{\pi}{2 \sqrt{2}} \]

Final Calculation

Double the result because of the symmetry adjustment previously made: \[ 2 \times \frac{\pi}{2 \sqrt{2}} = \frac{\pi}{\sqrt{2}} \]

Key concepts

Symmetry in integrals

Symmetry is a powerful tool in calculus, especially when dealing with integrals over symmetric limits. It can significantly simplify the evaluation process. Let's take a closer look at how symmetry works:
When a function is 'even', it means that it satisfies the property: \ f(x) = f(-x). In the context of integrals, if you have an integral of an even function over a symmetric interval such as \[-a, a\], you can simplify it by computing twice the integral from \[0, a\]:

• For example, \ \int_{-a}^{a} f(x)\, dx = 2 \ \int_{0}^{a} f(x) dx.

In the given problem, we identify that \1 + \cos^2(\phi)\ is even since \ \cos^2(\phi) = \ \cos^2(-\phi). Hence, the original integral can be transformed by adjusting the limits and halving the integral range, leading us to compute:

• \ \int_{-\frac{\pi}{4}}^{\frac{3\pi}{4}} \ \frac{d\phi}{\sqrt{1+\cos^2(\phi)}} = 2\ \ \int_{0}^{\frac{\pi}{4}} \ \frac{d\phi}{\sqrt{1+\cos^2(\phi)}}.
• This transformation simplifies the evaluation by focusing on \[0, \frac{\pi}{4}\] and then doubling it.

Elliptic integrals

Often, while dealing with integrals that arise from complicated functions, such as those involving square roots of trigonometric functions, elementary functions fail to offer antiderivatives. Here, special functions like elliptic integrals come into play. Elliptic integrals might appear daunting, but with proper tables and reference materials, they become manageable:
Elliptic integrals can broadly be categorized into three types: First kind, Second kind, and Third kind. The integral we encountered \(\int_0^{\frac{\pi}{4}} \frac{d\phi}{\sqrt{1+\cos^2(\phi)}}\) falls under the category of the elliptic integral of the first kind. These special tables provide exact values for integrals that involve combinations of square roots and trigonometric functions.

• For instance, given: \(\int_0^{\frac{\pi}{4}} \frac{d\phi}{\sqrt{1+\cos^2(\phi)}} = \frac{\pi}{2 \sqrt{2}}\),

we can use these results directly to simplify our calculations without getting bogged down in complex manual integration.

Integration techniques

Mastering various integration techniques is essential for tackling a wide array of integrals you might encounter. Here’s a succinct guide to some of the key techniques:

• Substitution Method: Used primarily when the integrand involves a composite function. You substitute a part of the integrand with a single variable to simplify.


• Integration by Parts: Useful when the integral is a product of two functions. It is based on the formula: \(\int u dv = uv - \int v du\).


• Partial Fraction Decomposition: A method used when the integrand is a rational function, breaking it down into simpler fractions.
  

In our exercise, the symmetry property of the integrand and the standard result using elliptic integrals helped streamline the calculation process.
Each technique has its own utility based on the type of functions involved, and choosing the correct one can significantly simplify the integration process.
Remember: The more techniques you become comfortable with, the better prepared you'll be to handle various integrals efficiently!