# Interesting but tricky integration problem

• grignard
In summary, the conversation discusses how to evaluate the given integral without using partial fraction expansion. The individual has attempted to split the integral into three separate integrals, but is unsure if this counts as partial fraction expansion. They also mention using the rule for integrating (dx/a^2+x^2), but are unsure how to apply it in this case. It is suggested to use polynomial division to simplify the integral before proceeding.

## Homework Statement

Evaluate the integral, (x^2+3x-8/x^2+16)dx, without using partial fraction expansion. Now this has really confused me. I know that partial fraction expansion is not viable because the mumerator has a higher power overall than the denominator

## The Attempt at a Solution

S x^2 /(x^2 + 16) dx + S 3x /(x^2 + 16) dx - S 8 /(x^2 + 16) dx

i have split it up like this, but i am not sure if this counts as a partial fracion expansion?

Next id use the rule, that the integral of (dx/a^2+x^2)=1/a tan(-1) x/a +c. But there are several x^2 on top, so how this does fit in.
It probably easier than it looks, any help would be much appreciaed. Thanks

Use polynomial division to divide x^2+3x-8 by x^2+16 first.

grignard said:
I know that partial fraction expansion is not viable because the mumerator has a higher power overall than the denominator

Minor point -- the numerator does NOT have a higher power than the denominator. The degree of each is 2.

## 1. What is an integration problem?

An integration problem is a mathematical problem that involves finding the integral of a given function. The integral represents the area under the curve of the function, and solving these problems requires knowledge of integration techniques such as substitution, integration by parts, and trigonometric substitution.

## 2. How do you approach an integration problem?

The first step in approaching an integration problem is to identify the type of function you are dealing with. This will help you choose the appropriate integration technique. Then, use the appropriate technique to solve the integral. It is also important to check your answer by differentiating it to ensure it is correct.

## 3. What makes an integration problem tricky?

An integration problem can be tricky for several reasons. It may involve complex functions, difficult substitution or integration by parts, or it may require multiple steps to solve. These problems also require a good understanding of integration techniques and the ability to apply them in different scenarios.

## 4. How can I improve my skills in solving integration problems?

Practice is key when it comes to improving your skills in solving integration problems. Start with simpler problems and gradually work your way up to more complex ones. It is also helpful to review the different integration techniques and their applications. Seeking help from a tutor or joining a study group can also be beneficial.

## 5. Are there any tips for solving tricky integration problems?

One tip for solving tricky integration problems is to break them down into smaller, more manageable parts. This can help simplify the problem and make it easier to solve. Another tip is to try different techniques and approaches until you find one that works. Additionally, it is important to carefully check your work and make sure you have not made any mistakes or overlooked any steps.