Calculus problem

[Samjohn]

Hello everyone: here is a problem I can't solve

Find all functions f which satisfy equation (integral f(x) dx )(integral dx/f(x))=-1

the problem that I'm facing is that I don't know how to tackle this problem like:
integral f(x) dx = f(x)^n+1 /n+1
integral dx/f(x) = ln |f(x)|
and they multiply both and I don't know the result to that.

I appreciate the help!

 

[DrPhil]

Find all functions f which satisfy equation (integrate f(x) dx )(integrate dx/f(x))=-1

\(\displaystyle \displaystyle \int f(x)\ dx \times \int \dfrac{dx}{f(x)} = -1 \)

Show us your work - what have you tried?

 

[Deleted member 4993]

Find all functions f which satisfy equation (integrate f(x) dx )(integrate dx/f(x))=-1

Please share your work with us .

If you are stuck at the beginning tell us and we'll start with the definitions

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

 

[DrPhil]

Hello everyone: here is a problem I can't solve

Find all functions f which satisfy equation (integral f(x) dx )(integral dx/f(x))=-1

the problem that I'm facing is that I don't know how to tackle this problem like:
integral f(x) dx = f(x)^n+1 /n+1 ...X
integral dx/f(x) = ln |f(x)|...X
and they multiply both and I don't know the result to that.

I appreciate the help!

You can't integrate directly as you have tried, BECAUSE you have "dx" instead of "d(f(x))" as the increment in the integrals. That is, you don't have a factor of f'(x) in either integrand. You will probably have to use integration by parts. I'll give that a try - but you have to check my work very carefully!

\(\displaystyle \displaystyle \int f(x)\ dx = x\ f(x) - \int x \ f'(x)\ dx \)

\(\displaystyle \displaystyle \int \dfrac{dx}{f(x)} = \dfrac{x}{f(x)} + \int \dfrac{x\ f'(x)}{f^2(x)}\ dx \)

I don't see a light at the end of the tunnel yet, but you can ask what properties of f(x) are required for the product of these two expressions to exist and have the value -1.