91Ó°ÊÓ

The technique of integration by substitution is akin to the reverse process of the chain rule of differentiation. It involves changing the variable of integration to simplify the integral and make it more manageable. The substitution technique is particularly helpful when you're dealing with composite functions or functions that are products of a function and its derivative.

To use this method, you first choose a new variable, typically denoted as \(u\), that represents a part of the original integrand. Then, you express the differential \(dx\) in terms of \(du\) and replace all occurrences of the original variable with \(u\) in the integrand. After carrying out the integral with respect to \(u\), you then substitute back the original variable to obtain the final result.

An example of this technique is apparent in the exercise, where setting \(u = \sqrt{x}\) simplifies the integration process. The next critical step is expressing \(dx\) as \(2u du\) and substituting these into the original integral. This transformation makes the integral manageable and allows us to integrate more easily.

Integrating radical expressions, which involve square roots or any root functions, can be challenging. These expressions are often simplified by substitution, allowing for the integrand to be expressed in a form that is directly integrable.

In the solved problem, the integrand contains the expression \( \sqrt{1-x}/\sqrt{x} \). By employing a clever substitution, the radical is assimilated into the new variable, \(u\), yielding a much simpler expression that is a function of \(u\). Standard integration techniques can then be applied.

It is also worth noting that, in general, whenever an integrand involves \( \sqrt{a^2 - u^2} \), \( \sqrt{a^2 + u^2} \), or \( \sqrt{u^2 - a^2} \), trigonometric substitution might be another potential approach. However, for our example, the substitution used was more straightforward and did not require such methods.

The arcsin function integration is a specific instance encountered when dealing with the integral of a radical expression following the pattern \( \sqrt{1-u^2} \). The result of such an integral actually corresponds to the inverse sine function, commonly referred to as arcsin or \(\becausein^{-1}\).

The formula used in the exercise for integrating \( \sqrt{1-u^2} \) is a standard result: \[ \int \sqrt{1-u^2} du = u \sin^{-1}(u) + \sqrt{1 - u^2} + C \.\] After using substitution to reach this form, we can directly apply this formula to find the integral. The arcsin function, or \(\becausein^{-1}\), is particularly important because it allows us to express the antiderivative in terms of elementary functions that are readily interpretable. Upon integrating and applying the arcsin function, the final step is to replace \(u\) with the original variable to express the answer in its original context, which completes the integration process.