91Ó°ÊÓ

Trigonometric substitution is a powerful technique used in calculus to solve integrals involving square roots, especially those with expressions such as \(\sqrt{a^2 + x^2}\), \(\sqrt{a^2 - x^2}\), or \(\sqrt{x^2 - a^2}\). This method takes advantage of the Pythagorean identities and trigonometric functions to simplify the integration process.

In our case, the integral \(\int \frac{1}{\sqrt{1+x^2}} \, dx\) suggests the use of tangent substitution. Here, we set \(x = \tan(\theta)\), which converts the expression to involve \(\sec(\theta)\), since \(\sqrt{1 + x^2} = \sec(\theta)\). This substitution simplifies the integral and transforms it into a more manageable form.

The matching identities required are:

• \(x = \tan(\theta)\)
• \(dx = \sec^2(\theta) \, d\theta\)
• \(\sqrt{1+x^2} = \sec(\theta)\)

These adjustments pave the way for the calculation of the integral by converting the complicated expression into simpler trigonometric functions, hence making the problem more solvable.

Definite integrals are a fundamental concept in calculus that calculate the net area under a curve within specified limits, providing a number rather than an expression. They extend the concept of antiderivatives to determine the actual value of a function across a certain interval.

In this exercise, we worked with a definite integral, \(\int_{1}^{e} \frac{dy}{y \sqrt{1+ (\ln y)^{2}}}\), which required changing the variables using substitution. Initially, we replaced \(y\) with \(\ln y\), transforming the initial bounds \(y=1\) and \(y=e\) into \(x=0\) and \(x=1\), respectively. This step is crucial as it sets the scene for integrating over the newly substituted variable.

After applying trigonometric substitution, we resolved the new integral \(\int_{0}^{1} \frac{1}{\sqrt{1+x^2}} \, dx\) and evaluated it between the bounds of 0 and 1. Calculating this definite integral gave us the final answer \(\ln(\sqrt{2} + 1)\). This process showcases the practical use of definite integrals in determining the precise value of original expressions within given limits.

Integration techniques are various methods in calculus that aid in solving integrals which are not straightforward. Each technique has its strategy and is applicable depending on the form of the function to be integrated.

For the given problem, we combined multiple methods to tackle the complex integral effectively.

• **Substitution Method:** Initially used to align the integral closer to a form manageable by trigonometric substitution. Here, \(x = \ln y\) was selected, simplifying our given expression substantially.
• **Trigonometric Substitution:** Employed to manage the square root within the integral confidently by transforming \(x = \tan(\theta)\), this took advantage of trigonometric identities like \(\sec^2(\theta)\) simplifying it to a standard integral of \(\sec(\theta)\).

By strategically combining these methods, the integration became significantly more feasible. Understanding when and how to apply these techniques is key in mastering calculus integration.