91Ó°ÊÓ

The change of variables is a technique used in calculus to simplify integrals by substituting a part of the integrand (the expression inside the integral) with a new variable. This substitution transforms the integral into a form that's easier to solve. In this exercise, we use the substitution

• Let: \( u = \sqrt{\tan x} \)

This substitution implies that \( \tan x \) can be expressed as \( u^2 \). By differentiating both sides, we find that

• \( \sec^2 x \, dx = 2u \, du \)

The choice of substitution is often guided by the aim of simplifying the expression considerably. The key is to find a substitution where the new integral is easier to evaluate. Here, the change in variables transforms a complex trigonometric function into a simple polynomial expression imposed on integrals. This method is instrumental when dealing with integrals involving composite functions.

Partial fractions involve decomposing a complex rational function into a sum of simpler rational expressions. This method greatly aids in integrating functions that are difficult to handle. After changing variables in the original problem to \( u = \sqrt{\tan x} \), we arrive at an integral that lends itself well to partial fraction decomposition:

• \( \frac{1}{2} \int (1 + u^4) \, du \)

Instead of tackling the entire complex fraction, the strategy here is to first break it down. Unfortunately, here it doesn't decompose further, but this is because the polynomial is already a straightforward form for integration. Essentials of this concept lie in the ability to recognize when a polynomial fraction can be expanded into simpler, more easily integrable parts. Overall, this approach makes calculus much more approachable, particularly in dealing with more involved expressions.

Definite integrals are a key tool used to calculate the accumulation of quantities, represented as the area under a curve between two specific points. In this exercise, we explore the definite integral

• \( \int_{0}^{\pi / 4} \sqrt{\tan x} \, dx \)

After performing the change of variables and integrating the new expression, we substitute back to the original variable and evaluate at the bounds, resulting in:

• \( \frac{3}{5} \)

The process involves substituting \( u = \sqrt{\tan x} \) and converting our bounds accordingly. Evaluating definite integrals often involves key calculus operations: substitution, differentiation, and simplification. This integral calculation understanding is crucial because it allows us to find exact values for areas or accumulations, providing precise results which are essential in different applied sciences and engineering tasks.