Irrational natural log integral

[mwaso]

Homework Statement

the indefinite integral of (1+lnx)^(1/2)/(xlnx) dx

Homework Equations

n/a

The Attempt at a Solution

There aren't any x^2 in the root sign, so I don't think it can be a trig substitution. The only logical u sub I see is to let u=lnx. In that case, du=dx/x so the integral becomes (u+1)^(1/2)/u du. Unfortunately, I don't have any good ways to evaluate that integral.

I wanted to try another sub letting m=(u+1)^(1/2) which makes m^2=u+1. Then u=m^2-1 and du=2mdm. that left me with 2 times the integral of m^2/(m^2-1) dm. I think I can solve that using partial fractions [integral of 1+1(m^2-1) dm : A(m-1)+B(m+1)=1, A=-1/2 and B=1/2, solution=m-1/2lnm+1/2lnm, backsub a few times to get (lnx+1)^(1/2)-(1/2)ln(lnx+1)+(1/2)ln(lnx+1)+c] but I'm not at all confident in that answer or in my methods there at the last.

 

[bdeln]

My first thought was the substitution u = xln(x), then du = (1 + ln(x)) dx, but the square root there messes it up.

 

[Dick]

Homework Statement

the indefinite integral of (1+lnx)^(1/2)/(xlnx) dx

Homework Equations

n/a

The Attempt at a Solution

There aren't any x^2 in the root sign, so I don't think it can be a trig substitution. The only logical u sub I see is to let u=lnx. In that case, du=dx/x so the integral becomes (u+1)^(1/2)/u du. Unfortunately, I don't have any good ways to evaluate that integral.

I wanted to try another sub letting m=(u+1)^(1/2) which makes m^2=u+1. Then u=m^2-1 and du=2mdm. that left me with 2 times the integral of m^2/(m^2-1) dm. I think I can solve that using partial fractions [integral of 1+1(m^2-1) dm : A(m-1)+B(m+1)=1, A=-1/2 and B=1/2, solution=m-1/2lnm+1/2lnm, backsub a few times to get (lnx+1)^(1/2)-(1/2)ln(lnx+1)+(1/2)ln(lnx+1)+c] but I'm not at all confident in that answer or in my methods there at the last.

Nice try. You've basically got it. Except that after you finished the m integration you substituted ln(x) for m in the log parts of the expression. m=sqrt(ln(x)+1). And don't forget your overall factor of 2.