- #1
Jude075
- 20
- 0
Homework Statement
Integrate (8-16x)/(8x^2-4x+1) dx
Q
Homework Equations
I separate it first 8/(8x^2-4x+1)dx -16x/(8x^2-4x+1)dx
Then I have no idea what to do next.
The Attempt at a Solution
none ;(
Jude075 said:Based on my calculator, I need to use trig substitution. But I have no clue:(
Ray Vickson said:Put away the calculator; it may be hurting you more than helping.
Here is a little hint: write ##8 - 16x = 4 + (4 - 16x)##; this will give you two separate terms that need to be integrated. For one of the terms, use the hint provided by the previous responder; for the other term, use partial fractions.
BTW: writing things as above is just a matter of practice and experience. After you have done lots of questions like this one, it will become second nature to you---but ONLY if you put away the calculator!
Thank you! With your help, I got -ln|8x^2-4x+1| +8arctan(x-1/4)+C by using separation , completing the square and trig substitution.Ray Vickson said:Put away the calculator; it may be hurting you more than helping.
Here is a little hint: write ##8 - 16x = 4 + (4 - 16x)##; this will give you two separate terms that need to be integrated. For one of the terms, use the hint provided by the previous responder; for the other term, use partial fractions.
BTW: writing things as above is just a matter of practice and experience. After you have done lots of questions like this one, it will become second nature to you---but ONLY if you put away the calculator!
Jude075 said:Thank you! With your help, I got -ln|8x^2-4x+1| +8arctan(x-1/4)+C by using separation , completing the square and trig substitution.
Hope this is the right answer:)
A rational function is a mathematical function that can be expressed as the quotient of two polynomial functions. It can also be written in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.
Integrating a rational function means finding the antiderivative or indefinite integral of the function. It involves finding a function whose derivative is equal to the given rational function.
To integrate a rational function, you first need to determine if the function can be expressed as a sum or difference of simpler rational functions. Then, you can use techniques such as partial fraction decomposition, substitution, or integration by parts to find the antiderivative.
Integrating rational functions is used in many fields such as physics, engineering, and economics. It can be used to solve problems involving motion, optimization, and area under a curve, among others.
Yes, there are some special cases when integrating rational functions. For example, if the function has a quadratic or higher order denominator, it may require more advanced techniques such as trigonometric substitution or use of partial fractions with repeated factors. Additionally, if the function is improper, meaning the degree of the numerator is greater than or equal to the degree of the denominator, then it may require further simplification before integration.