91Ó°ÊÓ

In Integral Calculus, the substitution method is a powerful technique used to simplify the process of integration. It involves replacing a given variable with another one to transform the integral into a more manageable form. In the original exercise, the integral \[ \int_{0}^{\pi / 4} \sqrt{\tan x} \sec ^{2} x \, dx \]is evaluated using the substitution method. Here, we consider the substitution \( u = \tan x \). This is chosen because its derivative \[ du = \sec^2 x \, dx \]matches part of the integrand. By substituting, the original complex expression becomes a simpler form, making the integration straightforward.

• Choose a substitution such that the derivative of the chosen substitution is present in the original integral.
• Adjust the integration limits according to the substitution.
• Transform the integral and evaluate it using the new variable.

Thanks to this method, we can focus on simpler integrals, making the task of finding solutions much less complex.

A definite integral is an important concept in calculus that calculates the area under a curve within a specified interval. In our exercise, we have:\[ \int_{0}^{\pi/4} \sqrt{\tan x} \sec^{2} x \, dx \]Here, we specifically find the area bounded by the curve from \( x = 0 \) to \( x = \pi/4 \). By following the substitution method and transforming the integral into:\[ \int_{0}^{1} \sqrt{u} \, du \]we identify that it's much easier to deal with. The act of evaluating a definite integral involves not only finding the antiderivative but also applying the limits to compute a numerical value. The fundamental theorem of calculus assures us that this process gives us accurate results when applied correctly.

• Identify the interval for the integration boundaries.
• Change these limits when using substitution.
• Evaluate the antiderivative and apply the new limits.

Finally, the integral value tells us the accumulated quantity, such as area or total change, within the given interval.

The concept of an antiderivative is foundational in solving integrals. Essentially, an antiderivative is a function whose derivative equals the original function we are integrating. To solve the definite integral in the exercise, knowing how to find the antiderivative was crucial. After substitution, the integral\[ \int \sqrt{u} \, du \]needed to be solved. We know that the antiderivative of \( \sqrt{u} \) or \( u^{1/2} \) is \[ \frac{2}{3} u^{3/2} \].The process of finding antiderivatives involves reversing differentiation. Thus, integral calculus can be seen as the reverse operation of differentiation.

• Identify the form of the function to integrate.
• Use known antiderivatives or integration techniques like substitution to find the solution.
• Plug in the limits in definite integrals to get the result.

For this exercise, knowing the antiderivative allowed us to compute the definite integral, ultimately giving us the result of \( \frac{2}{3} \).