How to find C in this equation?
$$\int_0^x f(t)~dt=\int_x^1t^2f(t)~dt + x^2/4 + x^4/8 +C$$
I attempted to move over $\int_0^x f(t)~dt$ to the right side, so that I could solve a definite integral for x, but the signs (it would become negative) don't work out.
$\endgroup$1 Answer
$\begingroup$First you should took the derivative of the equation to obtain $$ f(x) = -x^2f(x) + \frac{x}{2} + \frac{x^3}{2}, $$ so $f(x) = \frac{x+x^3}{2(1+x^2)}$. Now use that in the equation to obtain $$ \frac{x^2}{4} = \frac{1}{8} - \frac{x^4}{8} + \frac{x^2}{4} + \frac{x^4}{8} + C .$$ Then $C=-\frac{1}{8}$.
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