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## Homework Statement

[itex]\int\frac{dx}{x^{2}*\sqrt{6x+1}}[/itex]

## Homework Equations

## The Attempt at a Solution

t=sqrt(6x+1)

so x=t[itex]^{2}[/itex]-1

dx=2t dt

so new integral is

2*[itex]\int\frac{t}{(t^{2}-1)^{2}*t}[/itex]

the t's cancel and then i can turn the denominator into (t+1)[itex]^{2}[/itex]*(t-1)[itex]^{2}[/itex]

By partial fractions I get the new integral to be

(1/4)*( [itex]\int\frac{dt}{t+1}[/itex]+[itex]\int\frac{dt}{(t+1)^{2}}[/itex]-[itex]\int\frac{dt}{(t-1)}[/itex]+[itex]\int\frac{dt}{(t-1)^{2}}[/itex] )

so the first and third are easy. The second I make a substitution equal to t+1. The 4th I make a substitution equal to t-1.

After finding all the integrals I replace all the 't's with (sqrt(6x+1)) and my final answer is

(1/4)*((ln|[itex]\sqrt{6x+1}[/itex]+1|)-(ln|[itex]\sqrt{6x+1}[/itex]-1|)-[itex]\frac{1}{\sqrt{6x+1}+1}[/itex]-[itex]\frac{1}{\sqrt{6x+1}-1}[/itex])

But apparently it's wrong and after checking over it I can't find out where. I am getting super frustrated because even when I KNOW the procedure for a problem, I am more often wrong than right. Does anyone also have any general tips to prevent this?