Solve Improper Integral: $\int\frac{dx}{x\sqrt{x^2-4}}

[trajan22]

$$\int \frac{dx}{x \sqrt{x^2-4}}$$

there are bounds to this problem but it is irrelevant since my problem is with the integration and not finding the limits.

this integral resembles that of arcsec(x) but I am not sure how to deal with the -4.

is there any way to solve this with partial fractions? Or substitution, the squareroot in the denominator is throwing me off for some reason.

sorry for all the recent posts but I am trying to teach myself the second part of calculus.

thanks for the help

 

[cristo]

Try the substitution x=2sec(u)

 

[trajan22]

ok but why would i pick that as the substitution, what strategy did you use to choose that as a substitution

 

[cristo]

ok but why would i pick that as the substitution, what strategy did you use to choose that as a substitution

Well, look at the sqrt in the denominator. We know that sec²x-1=tan²x, and that the derivative of secx is secx*tanx. This hints towards the substitution x=secu. Since we want to be able to factor the 4 out of the sqrt (to enable us to use the trig identity above), we then take x=sqrt(4)secu=2secu.

 

[trajan22]

Sorry I've been sitting here trying to understand this but I'm still confused. How does this substitution help because if we make x=2secu then we have

$$\int \frac{dx}{(2sec(x))\sqrt{(2sec(x))^2-4}}$$

but how is this more manageable than the previous equation.

 

[d_leet]

Sorry I've been sitting here trying to understand this but I'm still confused. How does this substitution help because if we make x=2secu then we have

$$\int \frac{dx}{(2sec(u))\sqrt{(2sec(u))^2-4}}$$

but how is this more manageable than the previous equation.

I changed all your "x"s to "u"s in the integral because that is what they should be after the substitution, but you forgot to substitute for dx after making that substitution. Also it would help to simplify the part under the radical using a trigonometric identity.

 

[trajan22]

Right, sorry I was thinking of something else when i did that substitution. I think I understand this now, thanks for all the input.