# trig sub correct or incorrect

• Mar 4th 2014, 12:09 PM
Jonroberts74
trig sub correct or incorrect
$\displaystyle \int \frac{1}{x^2-6x+5}dx$ I complete the square for $\displaystyle \int \frac{1}{(x-3)^2-4}dx$ let (x-3) = 2 sec theta and dx = 2 sec theta tan theta dtheta

$\displaystyle \int \frac{2 sec\theta tan \theta d\theta}{4(sec^2 \theta -1}$ $\displaystyle \Rightarrow \frac{1}{2} \int \frac {sec \theta}{tan \theta} d\theta$

$\displaystyle \Rightarrow \frac{1}{2} \int csc \theta d\theta = \frac{1}{2} ln |csc \theta - cot \theta| + C$

$\displaystyle = \frac{1}{2} ln| \frac{x-3}{\sqrt{(x-3)^2 -4}} - \frac{2}{\sqrt{(x-3)^2 -4}}|+C$

is this correct? wolfram did it differently so I just want to check to see if my way is correct as well.
• Mar 4th 2014, 12:36 PM
romsek
Re: trig sub correct or incorrect
your way looks correct as well. The plots of your function and wolfram's integral lie on top of each other.

I have to say though that this is clearly a partial fractions problem and not a trig substitution problem even if you can find an substitution that works.

$x^2-6x+5=(x-5)(x-1)$ so a bit of work shows

$\dfrac{1}{x^2-6x+5}=\dfrac{1}{4}\left(\dfrac{1}{x-5}-\dfrac{1}{x-1}\right)$

and the integral pops right out.
• Mar 4th 2014, 12:50 PM
Jonroberts74
Re: trig sub correct or incorrect
ah that makes much more sense. its from a improper integral convergence or divergence question. thanks