Natural logarithm homework

1. Jun 16, 2008

heyman123

i have a problem plz help

Who can solve this for me?!?!?!

2. Jun 16, 2008

Staff: Mentor

We don't give out answers here on the PF. You need to show us your own work before we can offer tutorial help.

What are your thoughts on how to approach this problem?

3. Jun 16, 2008

heyman123

i can do it but i need the primitive of sin x and thats my problem, the rest of the problem i can solve , i just only cant manage to discover de primitive of sin x

4. Jun 16, 2008

Defennder

Is that a logarithm of base 10 or the natural logarithm?

5. Jun 16, 2008

Defennder

The indefinite integral doesn't appear to have an elementary solution. So I'm thinking there must be something special about the definite integral.

6. Jun 16, 2008

Dick

This is basically a trick. There is no elementary primitive. Here's a clue. Change the range of integration to 0 to pi/2 and call the integral I. Then you want -2*I. Now observe the integral of log(cos(x)) is also I. That's the clue. Add integral log(sin(x)) and log(cos(x)) and use a rule of logarithms and a trig identity and a u-substitution. Now you got an equation with a bunch of I's in it. Can you solve for I?

7. Jun 17, 2008

physixguru

Integrate by partial fraction.

8. Jun 18, 2008

rock.freak667

how come if you change from pi to pi/2, the integral is doubled? I get that if the integrand was just sinx but isn't lnsinx something entirely different?

9. Jun 18, 2008

Dick

ln(sin(x)) is just as symmetric as sin(x). The integral of it from 0->pi/2 is half the integral of it from 0->pi.

10. Jun 18, 2008

rock.freak667

oh ok then...I thought the graphs were very different.

11. Jun 18, 2008

Dick

They are very different. But they are still symmetric around x=pi/2. Did you solve the problem? It's really not that hard if you put your mind to it and know the secret hint. I only knew it because I've seen this problem before.

12. Jun 18, 2008

rock.freak667

I only solved it based on your hints but I didn't know it was symmetric at pi/2. But how you knew to change the limits of integration beats me

13. Jun 18, 2008

Dick

Because cos(x) is negative between pi/2 and pi. So log(cos(x)) isn't defined. It just seemed neater to restrict the range rather than put an absolute value in. That's all.

14. Jun 19, 2008

HallsofIvy

Staff Emeritus
Surely, you didn't get to where you are being expect to solve problems like this without learning that the derivative of cos x is -sin x??

15. Jun 20, 2008