Integration: Integrate x/((x^2-2x+5)^(1/2)) dx

[IgorB]

Hello all,

I'm with problems with the following integration:

Integrate x/((x^2-2x+5)^(1/2)) dx

I thought I have reached a solution, but it happens to be wrong, and I can't stop what I've done uncorrectly.
My solution is:

Integrate x/((x-1)^2+4)) dx
x = sec(a)
dx = tan(a)sec(a) da

tan^2(a) = sec^2(a) - 1
The new integral is:

Integrate tan(a)sec^2(a)/((tan^2(a)+4)^(1/2)) da

tan^2(a)+4 = b^2
2tan(a)sec^2(a) da = 2b db

The new integral:

Integrate b/(b^2)^(1/2) db = Integrate b/b db = Integrate db = b + C = (tan^2(a) + 4)^(1/2) +C = (sec^2(a) + 3)^(1/2) + C = (x^2 + 3)^(1/2) + C

But this is, for sure, wrong, as the answer should be:

(x^2 - 2 x + 5)^(1/2) + sinh^(-1)((x - 1)/2) + C
Please, help.

Thank you.


BR.,
Igor B.

 

[Deleted member 4993]

Hello all,

I'm with problems with the following integration:

Integrate x/((x^2-2x+5)^(1/2)) dx

I thought I have reached a solution, but it happens to be wrong, and I can't stop what I've done uncorrectly.
My solution is:

Integrate x/((x-1)^2+4)) dx
x = sec(a) → (x-1)^2 = [sec(a) - 1]^2 → \(\displaystyle \ne\)tan^2(a)
dx = tan(a)sec(a) da

tan^2(a) = sec^2(a) - 1
The new integral is:

Integrate tan(a)sec^2(a)/((tan^2(a)+4)^(1/2)) da .... Incorrect

tan^2(a)+4 = b^2
2tan(a)sec^2(a) da = 2b db

The new integral:

Integrate b/(b^2)^(1/2) db = Integrate b/b db = Integrate db = b + C = (tan^2(a) + 4)^(1/2) +C = (sec^2(a) + 3)^(1/2) + C = (x^2 + 3)^(1/2) + C

But this is, for sure, wrong, as the answer should be:

(x^2 - 2 x + 5)^(1/2) + sinh^(-1)((x - 1)/2) + C
Please, help.

Thank you.


BR.,
Igor B.

I would substitute as follows:

x - 1 = 2*tan(Θ) → dx = 2 * sec^2(Θ) dΘ

 

[IgorB]

I would substitute as follows:

x - 1 = 2*tan(Θ) → dx = 2 * sec^2(Θ) dΘ

Thank you for your reply, and for correcting me. I don't know how I couldn't see such mistakes. But still, I don't see how x - 1 = 2*tan(Θ) works actually. I've got stuck once again.

 

[stapel]

Thank you for your reply, and for correcting me. I don't know how I couldn't see such mistakes. But still, I don't see how x - 1 = 2*tan(Θ) works actually. I've got stuck once again.

Please reply showing how far you've gotten, so we can help you get un-stuck. Thank you!