Question

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers. $$ \int \frac{d x}{x^{1 / 2}+x^{1 / 3}} $$

Short answer

Use a CAS to evaluate \( \int \frac{dx}{x^{1/2}+x^{1/3}} \). Simplify and verify with tables.

Step-by-step solution

Set up the Integral

The integral we need to evaluate is \( \int \frac{d x}{x^{1/2} + x^{1/3}} \). To make it easier for evaluation, let's first express the terms with the same power base to simplify, if possible, before moving to computational tools.

Simplification of the Denominator

Rewriting the integral, we can factor out the smallest power, which is \( x^{1/3} \), from the denominator: \( x^{1/3}(x^{1/6} + 1) \). The integral now becomes \( \int \frac{x^{-1/3} \, dx}{x^{1/6} + 1} \), allowing for simplification after factoring.

Use of Computer Algebra System (CAS)

Input the simplified integral \( \int \frac{x^{-1/3} \, dx}{x^{1/6} + 1} \) into a CAS tool such as Wolfram Alpha, MATLAB, or any other available software to evaluate the actual integral explicitly.

Verification with Integration Tables

Check the results obtained from the CAS against standard integration tables if available. These tables often provide forms of common integrals that can be used for confirmation.

Key concepts

computer algebra system

A Computer Algebra System (CAS) is an advanced tool used to solve complex mathematical problems, including integrals. These systems can handle symbolic mathematics, allowing students and professionals to evaluate expressions without manually performing arduous calculations.
CAS tools come packed with features that simplify the process of solving integrals:

• They automate the execution of algebraic operations.
• They provide precise solutions for integrals and derivatives.
• They offer graphical representations of solutions for better understanding.

Popular CAS tools include software like Wolfram Alpha and MATLAB. These systems allow users to input complex expressions and receive an output almost instantaneously. For this exercise, using a CAS means students can bypass potential manual errors and directly progress to interpretation and application of results.

integration techniques

Integration techniques refer to the methods employed to solve integrals. Different techniques cater to different expressions, aiming to simplify the process of integration by breaking down complicated forms.
Some of the common integration techniques include:

• Substitution: This technique involves replacing a section of the integral with a single variable to make it simpler to solve.
• Integration by parts: Useful for integrals of products of functions, making use of the derivative-anti-derivative relationship.
• Partial fraction decomposition: Breaks down rational expressions into simpler parts for easier integration.

In this specific exercise, simplifying the base of the powers in the denominator before using a CAS is an example of preparing an integral for easier computational evaluation. Understanding and choosing the right technique is crucial for tackling complex integrals efficiently.

simplification of integrals

Simplification of integrals is a crucial step in solving them, as it often reduces complexity and can make integrating much more manageable. The first move is often to transform or factor the expression in question, as illustrated in this exercise.
Some approaches to simplification include:

• Factorization: As demonstrated, factorizing the denominator by finding a common term can simplify the evaluation process.
• Simplifying expressions: Transforming complex expressions into their simplest form makes subsequent steps easier.
• Combining like terms: This involves grouping similar powers or coefficients to refine the expression.

Simplification strategies can greatly influence the chosen integration technique, ultimately affecting the ease and speed of solving an integral. By pre-simplifying the expression, users prepare it for execution in a CAS, ensuring accurate and efficient computation.