Given that the continuous function $f: \Bbb R \longrightarrow \Bbb R$ satisfies $\int_0^\pi f(x) ~dx = \pi,$ Find the exact value of $\int_0^{\pi^{1/6}} x^5 f(x^6) ~dx.$
Let $g(t) = \int_t^{2t} \frac{x^2 + 1}{x + 1} ~dx.$ Find $g'(t)$.
For the first question: The way I understand this is that the area under $f(x)$ from $0$ to $\pi$ is $\pi$. Doesn't this mean that the function can be $f(x)=1$? Are there other functions that satisfy this definition? The second line in part one also confuses me, specifically the $x^6$ part!
For the second question: Does this have to do something with the Second Fundamental Theory of Calculus? I see that there are two variables, $x$ and $t$, that are involved in this equation.