Troubleshooting the Integral of dx/(x-sqrt(x+2)) in Calculus 2

[mathrocks]

This one integral has been given me some problems.

integral of dx/(x-sqrt(x+2))

please keep in mind I am only in calc 2.

 

[quantumdude]

This one integral has been given me some problems.

integral of dx/(x-sqrt(x+2))

please keep in mind I am only in calc 2.

You can do a u-substitution with u=(x+2)^1/2. Then du=(1/2)(x+2)^-1/2.

Now, you'll find that you don't have the du/dx in your integral, so you'll have to supply it yourself by multiplying the numerator and denominator of the integrand by 2(x+2)^1/2 and separating factors accordingly.

That will turn your integrand into a rational function of u which will be much easier to integrate.

 

[nizama]

Well first thing i always do is when i have any function given like this is to rationalize it ..meaning multiply all of this with (x+sqrt(x+2))/(x+sqrt(x+2))
You will get then this sqrt up...
(x+sqrt(x+2)) / ((x^2)-x-2)
then you have (x+sqrt(x+2)) / (x-2)(x+1)
Then you can simply separate this integral into two.. int_x / ((x-2)(x+1))dx + int_sqrt(x+2) / ((x-2)(x+1))dx
from there by substitution you can easily sole both of them