91Ó°ÊÓ

Understanding integration techniques is essential for solving complex calculus problems. These techniques provide strategies to evaluate integrals, especially when the integrand is not straightforward. In this exercise, we are tasked with integrating a function that, at first glance, looks complex: \[\int \frac{1}{\left[(x-1)^{3}(x+2)^{5}\right]^{1 / 4}} \ d x\]One common technique for solving such problems is by simplifying the integrand. You can often rewrite complex expressions into simpler forms. For this exercise, recognizing that the expression \[[(x-1)^3(x+2)^5]^{-1/4}\] can be simplified into \[(x-1)^{-3/4}(x+2)^{-5/4}\]allows us to apply further techniques such as substitution. These manipulations are often necessary to transform the integral into a form where standard integration formulas or techniques can be applied effectively.

The substitution method is a powerful tool in calculus, specifically useful for dealing with integrals that are not readily solvable. This technique involves changing the variables in the integrand to simplify the integral. In our exercise: \[\int (x-1)^{-3/4}(x+2)^{-5/4} \, d x,\] we use the substitution: \[u = \frac{x-1}{x+2}\]This choice aims to simplify the relationship between the variables involved. By substituting, we derive: \[u = \frac{x-1}{x+2},\]and solve for \(x\) as:\[x = \frac{u+1}{1-u}\]Differentiating gives us:\[dx = \frac{3}{(1-u)^2} du\] This transformation is crucial as it turns the integral into one involving relatively simpler terms in \(u\), which can then be tackled with basic integration techniques. Converting the entire integrand into the new variable \(u\) allows us to solve the integral more easily.

Solving calculus problems involves a systematic approach that requires understanding and application of various techniques and ideas. For our given problem, which involves evaluating the integral:\[\int \frac{1}{\left[(x-1)^{3}(x+2)^{5}\right]^{1 / 4}} \ d x,\] we make use of multiple problem-solving strategies.**Breaking Down the Problem**- Analyze the integrand's structure to understand potential simplification methods.- Use substitution to transform the integral into an easier one by expressing it in terms of another variable.- Simplify the integral based on known forms or patterns that match standard integral solutions.By treating the integral with substitution and simplification techniques, we ultimately evaluate it to:\[\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{1/4} + c,\] which corresponds to option (A) given in the problem. Developing these skills in structured problem solving and the ability to recognize when and how to apply integration techniques is crucial for becoming proficient in calculus.