I'm not sure how to approach this question, but I know it has something to do with taking the derivative of this. Can anyone please help me out?
Let x > 0. Prove that the value of the following expression does not depend on x: $$\int_{0}^{x}\frac{1}{1+t^4} dt + \frac{1}{3}\int_{0}^\frac{1}{x^3} \frac{1}{1+t^\frac{4}{3}} dt$$
Let $$f(x) := \int_0^x \frac{1}{1+t^4} dt + \frac{1}{3}\int_0^{\frac{1}{x^3}} \frac{1}{1+t^{\frac{4}{3}}} dt$$ Then differentiating, $$f'(x) = \frac{1}{1+x^4} + \frac{1}{3} \left( \frac{1}{1+x^{-4}} \cdot \frac{-3}{x^4} \right) = \frac{1}{1+x^4} - \left( \frac{1}{x^{4} + 1} \right) = 0$$ Hence, since $f'(x) = 0$ for all $x$, $f$ must be constant, so it does not depend on $x$.