Tricky Integration

1. Dec 5, 2017

squenshl

1. The problem statement, all variables and given/known data
How do I solve $\frac{20\ln{(t)}}{t}$???

2. Relevant equations

3. The attempt at a solution
Is it easier to calculate this without integrating by parts???
I'm not sure where to start.

2. Dec 5, 2017

andrewkirk

Solving by integration by parts is easy - very few lines. I can't think of an easier way.

3. Dec 5, 2017

Tallus Bryne

There is an easier way. Try a simple "u-substitution".

4. Dec 5, 2017

Staff: Mentor

You have $f\cdot f'$ which is half of $(f^2)'$. Done.

5. Dec 5, 2017

Tallus Bryne

I like the simplicity, but shouldn't it be just half of $(f^2)$ ?

edited: forgot the "half of"

6. Dec 5, 2017

Staff: Mentor

$f^2$ is the solution after integration, but I said $f \cdot f' = \frac{1}{2} \cdot (f^2)'$ for the integrand. Of course this all is a bit sloppy: no integration boundaries or the constant, no mentioning of $\ln |x|$ in the OP and no $dt 's$.

7. Dec 5, 2017

Tallus Bryne

Agreed. Thanks for clearing that up.

8. Dec 6, 2017

Math_QED

Substitute $u = \ln t, du = \frac {1}{t}dt$

9. Dec 6, 2017

squenshl

Thanks everyone. The solution is $10\ln{(t)}^2$.

10. Dec 7, 2017

SammyS

Staff Emeritus
You might like to remove an ambiguity.

Is that the natural log of the square of t, or is it the square of the natural log of t ?

Also, I suspect that you need to include a constant of integration .

11. Dec 7, 2017

MidgetDwarf

Typically for integrals involving ln in a typical calculus course...

Always try u sub. If that does not work, then integration by parts. Keep this in mind.