The problem involves integrating the rational function (8-16x)/(8x^2-4x+1) with respect to x. Participants are exploring different methods to approach the integration, including separation of terms and potential substitutions.
The discussion is ongoing, with participants providing hints and suggestions for breaking down the integral. There is a mix of approaches being explored, including separation of terms and the use of trigonometric substitution. Some participants express confusion about specific methods, indicating that the conversation is still developing without a clear consensus.
Participants are navigating the complexities of integration techniques and are encouraged to reconsider their reliance on calculators. There is mention of specific algebraic manipulations, such as completing the square, which may influence the integration process.
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:)