# Solving a Tricky Integral Homework Problem

• iatnogpitw
In summary, the problem is to integrate ((x^3) - 1) / ((x^3) + x) dx, using partial fractions or polynomial division.
iatnogpitw

## Homework Statement

Okay, here's the problem: integrate ((x^3) - 1) / ((x^3) + x) dx

## Homework Equations

What is the solution and the method?

## The Attempt at a Solution

I know this involves partial fractions, which I attempted to no avail, since the numerator is the same degree as the denominator. I even factored the numerator into (x - 1)((x^2) + x + 1), but I can't seem to figure out what to do.

$$\int \frac{x^3 - 1}{x^3 + x}\mathrm{d}x$$
If factoring doesn't help, I'd try splitting it into a sum of two integrals.

If you're really stuck, mouseover this for a clue:
x^3 - 1 = (x^3 + x) - (x + 1)

Last edited:
Or you can simply divide the numerator by the denominator using polynomial division.

## 1. How do I determine the appropriate integration method for a tricky integral problem?

There are several methods for solving integrals, such as substitution, integration by parts, and partial fractions. To determine the appropriate method, you should first try to simplify the integral by factoring or using algebraic manipulation. If the integral still appears challenging, consider using a substitution to transform the integral into a more manageable form. If the integral involves products or powers of functions, integration by parts may be a useful method. Finally, if the integral contains fractions, partial fractions may be the most effective approach.

## 2. What should I do if I am stuck on a tricky integral problem?

If you are stuck on a tricky integral problem, don't give up! Take a break and come back to it with a fresh perspective. You can also try looking for online resources or asking a classmate or instructor for help. Additionally, don't be afraid to try different integration techniques or approaches to see if they lead to a solution. Sometimes, it may be helpful to rewrite the integral in a different form or use properties of integrals to simplify it.

## 3. How can I check my answer to a tricky integral problem?

One way to check your answer to a tricky integral problem is to use a graphing calculator or software to graph the original function and the antiderivative. If the graphs match, it is likely that your answer is correct. You can also take the derivative of your antiderivative and see if it gives you the original function. Another method is to use numerical integration, such as the trapezoidal rule or Simpson's rule, to approximate the value of the integral and compare it to your answer.

## 4. What are common mistakes to avoid when solving a tricky integral problem?

Some common mistakes to avoid when solving a tricky integral problem include forgetting to include the constant of integration, making algebraic errors while simplifying the integral, and missing necessary substitutions or applying integration techniques incorrectly. It is also important to carefully check your work and make sure that your answer makes sense in the context of the original problem.

## 5. How can I improve my skills in solving tricky integrals?

The best way to improve your skills in solving tricky integrals is through practice. Make sure to review the fundamental concepts and techniques of integration, such as the power rule, substitution, and integration by parts. You can also challenge yourself by attempting more challenging problems or seeking out additional resources, such as online tutorials or textbooks. Additionally, seeking help from a tutor or attending a study group can also help you improve your understanding and problem-solving abilities.

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