91Ó°ÊÓ

The substitution method is a technique used in integration to simplify complex integrals by changing variables. In this process, we replace a variable with another that simplifies the integral, making it easier to solve.

In this exercise, we began with the integral \( \int \frac{d x}{x^{4}+x^{2}+1} \). The substitution \( u = x^2 \) was proposed because the denominator could be expressed simply in terms of \( u \). This helped in transforming the integrand from a function of \( x \) to a function of \( u \).

To carry out the substitution, we calculated the differential \( du \) in terms of \( dx \). This allowed us to express the entire integral in terms of \( u \), leading to the simpler integral \( \frac{1}{2} \int \frac{d u}{u^2 + u + 1} \).

Key points when using the substitution method:

• Select a substitution that simplifies the integral significantly.
• Find the differential of the new variable relative to the old variable.
• Convert all expressions, including boundaries if they exist, in terms of the new variable.
• Don't forget to reverse the substitution at the end to express the solution in terms of the original variable.

This first step laid the groundwork for simplifying and eventually solving the given integral.

Trigonometric substitution is a technique particularly useful when an integrand involves expressions like \( \sqrt{a^2 + x^2} \), \( \sqrt{a^2 - x^2} \), or \( \sqrt{x^2 - a^2} \). By substituting a trigonometric function of a new variable, the integral simplifies into a form often involving basic trigonometric identities.

In the presented solution, the expression inside the integral assumes the form \( (u + \frac{1}{2})^2 + \frac{3}{4} \), equivalent to \( a^2 + x^2 \). Hence, trigonometric substitution was used with \( u + \frac{1}{2} = \frac{\sqrt{3}}{2} \tan(\theta) \) and \( a^2 = \frac{3}{4} \).

Here are the steps involved:

• Identify the substitution based on the format related to the trigonometric identity.
• Express \( du \) in terms of \( \theta \) variable substitutions, like \( du = \frac{\sqrt{3}}{2} \sec^2(\theta)d\theta \) in this case.
• Simplify the integral using trigonometric identities, for instance, \( 1 + \tan^2(\theta) = \sec^2(\theta) \).
• Once simplified to a basic trigonometric form, integrating becomes straightforward, often resulting in \( \theta \) form solutions.

This method significantly simplified our integral here, converting it into a basic trigonometric integral.

Completing the square is a method used to express a quadratic expression as a perfect square plus or minus a constant. This technique is highly useful for recollecting patterns in integrals or for solving quadratic equations.

In this integral, we started by expressing \( u^2 + u + 1 \) in a transformed form. By completing the square, it was rewritten as \( (u + \frac{1}{2})^2 + \frac{3}{4} \).

The steps to complete the square are as follows:

• Take the coefficient of the linear term, \( u \) in this case, divide it by 2, and then square it. Here, \( \frac{1}{2} \) is squared to get \( \frac{1}{4} \).
• Add and subtract this squared number inside the expression \( u^2 + u + 1 \).
• Factor the perfect square trinomial. Convert the expression into \( (u + \frac{1}{2})^2 + \frac{3}{4} \).

By completing the square, the quadratic form becomes suitable for trigonometric substitution, thus paving the way for evaluation of more complex integrals. This method is a powerful tool in integration when dealing with quadratics inside denominators or square roots.