91Ó°ÊÓ

In calculus, the indefinite integral is essentially the antiderivative of a function, used to construct functions that can calculate the area under the curve of the original function. It is represented by the integral sign followed by the function and the differential, for example, \( \int f(x)\, dx \). The result of an indefinite integral includes the family of all antiderivatives plus a constant of integration, typically denoted as \( C \).

Let’s understand the indefinite integral in the context of our exercise. We were asked to integrate \( \frac{1}{x^2 - 6x + 34} \) with respect to \( x \). Here, we were not given any specific bounds, which means we were working with an indefinite integral. After applying the method of completing the square and some substitution, we simplified the integral to find its antiderivative, which yields a general formula for the area under the curve of the rational function \( \frac{1}{x^2 - 6x + 34} \).

In practice, when finding the antiderivative, we look for a function whose derivative returns the original function under the integral sign. For our integral, the antiderivative in terms of \( u \) was \( \frac{1}{5}u \), leading to the general solution \( \frac{1}{5}\arctan{\frac{x-3}{5}} + C \) when we substituted back in terms of \( x \).

The technique of trigonometric substitution is used in integration to simplify the integrals that contain square roots of quadratic expressions or, as in our case, to transform the integral into a more familiar trigonometric form. This substitution is based on the Pythagorean identity, which connects trigonometric functions with quadratic expressions. For instance, the identity \( 1 + \tan^2{u} = \sec^2{u} \) comes in handy.

In the step-by-step solution, after completing the square, we noticed that the denominator of our integrand could be associated with a Pythagorean identity by the substitution \( x - 3 = 5\tan{u} \), leading to a new trigonometric expression. This method transforms the integral into one involving standard trigonometric functions, which are often easier to integrate. After the substitution, the integral became a simpler expression in terms of \( u \), which we could then integrate directly.

The method of integration by substitution, also known as u-substitution, is the reverse process of the chain rule in differentiation and is used to find integrals of more complicated functions. By choosing a new variable \( u \) to represent a part of the original integrand, it often simplifies the integral into a form that is easier to evaluate.

When applying integration by substitution to solve an integral, we aim to substitute a part of the integrand with a new variable \( u \) and express \( dx \) in terms of \( du \), which then allows us to rewrite the integral in a simpler form. For our example, the substitution \( x - 3 = 5\tan{u} \), and subsequently \( dx = 5\sec^2{u}\, du \), allowed us to transform the original integral into a much simpler form that we could evaluate easily.