91Ó°ÊÓ

The substitution method is a powerful technique for solving integrals where you replace a complex part of the integrand with a single variable, to simplify the integration process. For the given exercise, we use substitution to tackle the integral \(\int \frac{x}{x^{2}+4} dx\).To start, identify a substitution that can simplify the expression. Here, we set \(u = x^2 + 4\), transforming the integral into a simpler form. Correspondingly, the differential \(du\) is derived as \(du = 2x\,dx\), or rearranged to \(\frac{du}{2} = x\,dx\). This allows transformation of the integrand to \(\frac{1}{u}\cdot\frac{du}{2}\).

• Simplify by integrating \(\frac{1}{u}\) to obtain \(\frac{1}{2}\ln|u| + C\).
• Finally, reverse the substitution by replacing \(u\) with \(x^2 + 4\), yielding \(\frac{1}{2}\ln(x^2+4) + C\).

This method turns a seemingly complicated integral into a much simpler expression by transforming the variable and narrowing it down to basic logarithmic integration. It's efficient for cases where the derivative of the chosen \(u\) term is straightforward.

Trigonometric substitution is another fascinating method used for solving integrals that involve expressions of the form \(\sqrt{a^2+x^2}\), \(\sqrt{x^2-a^2}\), or \(\sqrt{a^2-x^2}\).In the exercise, using trigonometric substitution involves replacing \(x\) with \(2\tan\theta\). Consequently, \(dx\) becomes \(2\sec^2\theta\,d\theta\) and \(x^2 + 4\) matches \(4\sec^2\theta\). This substitution turns the integrand \(\frac{x}{x^2+4}\) into \(\tan\theta\). Now, the integral becomes \(\int \tan\theta\,d\theta\).

• The integration of \(\tan\theta\) yields \(-\ln|\cos\theta| + C\).
• To revert back to the original variable \(x\), use the trigonometric identity \(\cos\theta = \frac{2}{\sqrt{x^2+4}}\).
• The resulting expression is \(\ln|\sqrt{x^2+4}| + C\), simplifying further to \(\frac{1}{2}\ln(x^2+4) + C\).

Trigonometric substitution is highly effective for working with integrals containing square roots, and while it might seem unfamiliar at first, it becomes an insightful tool once grasped.

While the exercise mainly addresses indefinite integrals, understanding the concept of definite integrals is crucial for comprehensive learning.Definite integrals differ as they evaluate the area under a curve between two specified points, \(a\) and \(b\). For definiteness in our integral, you would calculate \[\int_a^b \frac{x}{x^2+4} \, dx\] over a particular interval rather than simply finding the antiderivative.

• Begin by following similar substitution techniques as for indefinite integrals.
• After integration, replace the variable back and evaluate the limits \(a\) and \(b\) in the expression.
• This gives you a numerical value representing the area under the curve defined by the integrand between \(x = a\) and \(x = b\).

Definite integrals provide not just the antiderivative form, but a specific value that actually quantifies certain aspects, like area or total change. Understanding the transition from indefinite to definite integrals is foundational in calculus.