How Do You Solve for f(x) in a Trigonometric Integral?

[redtree]

Suppose y = f(x) * sin (kx), where k = wavenumber.

If $$\int y*dy$$ = 3*kx, solve for f(x)

 

[tiny-tim]

Hi redtree!

Suppose y = f(x) * sin (kx), where k = wavenumber.

If $$\int y*dy$$ = 3*kx, solve for f(x)

erm … $$\int y*dy\ =\ \frac{1}{2}\,y^2$$

do you mean $$\int_0^x y(z)*dz\ =\ 3\,kx$$ ?

If so, just differentatiate both sides.

 

[HallsofIvy]

If you really do mean $\int y dy= 3kx$, then (1/2)y²+ C= 3kx so
$$y= f(x)sin(kx)= \sqrt{6kx- 2C}$$
and
$$f(x)= \frac{\sqrt{6kx- 2C}}{sin(kx)}$$
where C and be any constant.

But I suspect tiny-tim is right.