Thread: Integration Help

I need to integrate 1/(16x^3-4x^2+4x-1)

I first split the denominator into (4x-1)(4x^2+1) and used partial fraction decomposition to get:

the integral of (4/5)/(4x-1)+((-4/5)x-(1/5))/(4x^2+1)

I know how to solve the first segment of the integral, but I don't know how to integrate ((-4/5)x-(1/5))/(4x^2+1)

Originally Posted by Simon777

I need to integrate 1/(16x^3-4x^2+4x-1)

I first split the denominator into (4x-1)(4x^2+1) and used partial fraction decomposition to get:

the integral of (4/5)/(4x-1)+((-4/5)x-(1/5))/(4x^2+1)

I know how to solve the first segment of the integral, but I don't know how to integrate ((-4/5)x-(1/5))/(4x^2+1)

Try splitting it into two rational expressions - the first with (-4/5)x in the numerator and the second with -(1/5) in the numerator.
The first one will involve ln(4x^2+1),
the second will need substitution.

So you have:

Originally Posted by ANDS!

So you have:

With everything after the subtraction sign, why is it just (x^2+1)? Shouldn't it be (4x^2+1)? That was what the original problem was split into.

Originally Posted by Debsta

Try splitting it into two rational expressions - the first with (-4/5)x in the numerator and the second with -(1/5) in the numerator.
The first one will involve ln(4x^2+1),
the second will need substitution.

Thank you, I got a little farther with that advise, but what is the integral of:

1/(4x^2+1)

I can't do u substitution and I can't convert this into tan^-1 because of the 4.

Originally Posted by Simon777

Thank you, I got a little farther with that advise, but what is the integral of:

1/(4x^2+1)

I can't do u substitution and I can't convert this into tan^-1 because of the 4.

edit: mistake. refer to skeeter's post

Originally Posted by Simon777

Thank you, I got a little farther with that advise, but what is the integral of:

1/(4x^2+1)

I can't do u substitution and I can't convert this into tan^-1 because of the 4.

Originally Posted by Simon777

With everything after the subtraction sign, why is it just (x^2+1)? Shouldn't it be (4x^2+1)? That was what the original problem was split into.

It is. Doesn't change using trig sub though. Even if it wasn't a perfect square.

Thank you all so much for the help. Can someone please check my answer?

ln|((4x-1)^(1/5)) / ((4x^2+1)^(1/10))| - (1/10)arctan(2x) + C

Originally Posted by Simon777

Thank you all so much for the help. Can someone please check my answer?

ln|((4x-1)^(1/5)) / ((4x^2+1)^(1/10))| - (1/10)arctan(2x) + C

Yes that looks right. Just one thing, because of the yukky exponents in the first term, I would have written the first term as:
(1/5) ln|(4x-1)/sqr(4x^2+1)|

Originally Posted by Debsta

Yes that looks right. Just one thing, because of the yukky exponents in the first term, I would have written the first term as:
(1/5) ln|(4x-1)/sqr(4x^2+1)|

Thanks for taking the time to check that.