Antiderivative Help: Calculate Integral

[OmniNewton]

Homework Statement

Calculate the following integral:

Homework Equations

N/A

The Attempt at a Solution

from this point I tried a u-substitution by letting u = -3 + 4/x -1/x^2 but this seemed to fail.

Are any suggestions possible?

 

[SteamKing]

Homework Statement

Calculate the following integral:

Homework Equations

N/A

The Attempt at a Solution

from this point I tried a u-substitution by letting u = -3 + 4/x -1/x^2 but this seemed to fail.

Are any suggestions possible?

Try factoring the quadratic under the root sign first. You should be able to find two linear factors.

 

[OmniNewton]

Try factoring the quadratic under the root sign first. You should be able to find two linear factors.

Alright thank you will attempt that now.

 

[OmniNewton]

Alright thank you will attempt that now.

Alright I have the indefinite integral factored for the argument however I am unsure how to proceed.

I'm going to let u = x-1 which means x = u + 1 and for the second factor (3x-1) = (3x - 1 - 2 + 2) = [3(x-1) + 2]

 

[OmniNewton]

Alright I have the indefinite integral factored for the argument however I am unsure how to proceed.

I'm going to let u = x-1 which means x = u + 1 and for the second factor (3x-1) = (3x - 1 - 2 + 2) = [3(x-1) + 2]

I may not be approaching this the right direction since after attempting this method I seem to be unable to proceed

 

[Incand]

There's usually several substitutions that work for these types of problems. The simplest one seems to be $t=1/x$. At least that seem to work for me (but I did this rather quickly so I may have done something wrong).

Another approach that should work would be is to complete the square and then use the standard substitution $\arcsin$ but in this case it seems to be rather complicated and you would need another substitution $s=\tan t/2$ after as well.

 

[azagaros]

That looks like I would complete the square of the term under the square root to get something to match arcsec or arccsc. 1/(u √ (u²-a²) is the one I believe i am thinking of..