Define all functions using the main statement

Question

Define all functions that are continious and fullfill the equation

$$ f(x) = -1 + \int_0^{x^2} \frac{(f(\sqrt{t})^2 \sin t}{\cos^2t} dt$$

I'm completely lost on this one. I think that you should use:

$$ S(x) = \int_a^x f(t) dt $$ in some way.

Answer 1

iveqy

Using fundamental theorem of calculus we get: $~f'(x)=\frac{f^2(x)sinx^2}{cos^2{x^2}}2x$. This yields $\int{f^{-2}}df=\int{}tanx^2secx^2xdx~$. So, we have, $\frac{-1}{f(x)}=secx^2+c$ . But $~f(0)=-1$. Therefore,$~c=0$ and $f(x)=-cosx^2$.