QIntegrate rational function: How do I solve (8-16x)/(8x^2-4x+1)dx?

[Jude075]

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 ;(

 

[SteamKing]

Separation won't work in this case.

Look at the numerator. Isn't that expression close to the derivative of the denominator? See if you can work out the integral based on this line.

 

[Jude075]

Based on my calculator, I need to use trig substitution. But I have no clue:(

 

[Ray Vickson]

Based on my calculator, I need to use trig substitution. But I have no clue:(

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]

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!

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.
Hope this is the right answer:)

 

[Dick]

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:)

No, I don't think the arctan part is correct. Can you show your work?