The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $$\int_{0}^{\infty} \frac{\cos x d x}{\left(1+9 x^{2}\right)^{2}}$$

Complex analysis is a branch of mathematics that studies functions of complex numbers. Unlike real analysis, complex analysis deals with complex functions, which are functions that have complex numbers as their variables and outputs. It offers powerful techniques for solving integrals, especially those difficult to handle using real-variable methods.
In the context of our exercise, we translate a real integral into a complex one. This step lets us use advanced tools like the residue theorem. By considering complex functions, we make the integral easier to work with and often uncover deeper structures in the problem.

• Complex Plane: Just like the real number line, but for complex numbers.
• Complex Functions: Functions that take and return complex numbers.
• Contours and Paths: Curves along which we integrate in the complex plane.

Understanding these basics improves our ability to handle integrals effectively, paving the way for solving more intricate mathematical problems.

The residue theorem is a fundamental tool in complex analysis. It allows us to calculate complex integrals via residues at singular points—where the function behaves 'badly.' This theorem is incredibly effective for problems like ours.
In simpler terms, residues can be thought of as the 'weights' located at singular points, which drastically simplify the integration process. To use the residue theorem:

• Identify the singularities of the integrand inside the contour.
• Calculate the residues at these singularities.
• Sum up these residues and multiply by \(\ 2\pi i \) to find the integral.

For our given integral, we convert it into a complex contour integral, locate singularities in the complex plane, compute the residues, and use the theorem to find the integral efficiently.

An improper integral is an integral where either the interval of integration is infinite or the function has infinite discontinuities within that interval. Both types require special care when evaluating.
In our problem, \( \int_{0}^{\infty}\ \frac{\cos x d x}{(1+9x^{2})^{2}} \), the interval is from 0 to infinity, making it an improper integral. These are often easier to handle using techniques from complex analysis. Key steps involved are:

• Rewriting the integral in terms of a complex function.
• Using contour integration or other methods to evaluate it.
• Relating the result back to the real number context.

By using complex integration methods, we can handle improper integrals more gracefully and with greater insight.

The complex exponential form is an excellent tool for simplifying many types of integrals and differential equations. It uses Euler's formula: \ e^{ix} = \cos x + i \sin x \.
This form is especially useful for converting trigonometric functions into exponential ones, simplifying integration. For our integral, using \( e^{ix} \) instead of \ \cos x \ lets us to switch from a real to complex setting, thus making the integral more straightforward to deal with.
To apply this:

• Replace trigonometric functions with their exponential equivalents.
• Reformulate the integral in terms of complex exponentials.
• Use standard techniques in complex analysis to solve the integral.

Using complex exponentials in such ways not only simplifies the calculation process but also brings out the underlying harmony in mathematical expressions.