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 May 20th, 2013, 06:57 PM #1 Member   Joined: Apr 2012 Posts: 72 Thanks: 3 What is the final solution and the way to solve the integral
 May 20th, 2013, 08:05 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: What is the final solution and the way to solve the inte We are given: $\int\frac{1}{x\ln(x)\ln(\ln(x))\ln(\ln(\ln(x)))\ln (\ln(\ln(\ln(x))))\ln(\ln(\ln(\ln(\ln(x)))))}\,dx$ Let: $u_1=\ln(x)\,\therefore\,du_1=\frac{1}{x}\,dx$ and we have: $\int\frac{1}{u_1\ln(u_1)\ln(\ln(u_1))\ln(\ln(\ln(u _1)))\ln(\ln(\ln(\ln(u_1))))}\,du_1$ Now, continue substituting in the same way, and you should see a pattern develop...
 May 20th, 2013, 08:11 PM #3 Member   Joined: Apr 2012 Posts: 72 Thanks: 3 Re: What is the final solution and the way to solve the inte Thanks! Is there any trick or shortcut?
 May 20th, 2013, 08:19 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: What is the final solution and the way to solve the inte Once you see the pattern (which you can see even from the first substitution) you will know the result without having to go through then entire process...
 May 20th, 2013, 08:52 PM #5 Member   Joined: Apr 2012 Posts: 72 Thanks: 3 Re: What is the final solution and the way to solve the inte Sorry but I could not solve it. Could you please show the simplest way?
 May 20th, 2013, 09:08 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: What is the final solution and the way to solve the inte Let's look first at: $\int\frac{1}{x}\,dx=\ln|x|+C$ Now, consider: $\int\frac{1}{x\ln(x)}\,dx$ Let $u=\ln(x)\,\therefore\,du=\frac{1}{x}\,dx$ and we have: $\int\frac{1}{u}\,du=\ln|u|+C=\ln|\ln(x)|+C$ Next, consider: $\int\frac{1}{x\ln(x)\ln(\ln(x))}\,dx$ Let $u_1=\ln(x)\,\therefore\,du_1=\frac{1}{x}\,dx$ and we have: $\int\frac{1}{u_1\ln(u_1)}\,du_1$ Now we know from the previous integral we may write: $\int\frac{1}{u_1\ln(u_1)}\,du_1=\ln|\ln(u_1)|+C=\l n|\ln(\ln(x))|+C$ Do you see the pattern?

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