We are given the following expression :
$$F_1(t) = \int_{0}^{\pi}{\frac{\cos(x)dx}{(1+t\cos(x))^2}}$$ $$F_2(t) = \int_{0}^{1}{\frac{\log(1+tx)}{1+x^2}}dx$$
Show the existence and evaluate $F_1'(t)$ and $F_2'(t)$
I don't know how I can prove the existence of both $F_1'(t)$ and $F_2'(t)$.
To evaluate $F_1'(t)$ and $F_2'(t)$, I thought that using the partial fraction decomposition method could help us but I'm not sure.
Any help would be a lot appreciated, thanks in advance.
As the integrand is $\mathcal{C}^1([0,1])$, if $t>-1$ we can use the Leibniz integral rule and get $$F'_2(t)=\int_0^1 \frac{\partial \frac{\log (t x+1)}{x^2+1}}{\partial t} \, dx=\int_0^1 \frac{x}{\left(x^2+1\right) (t x+1)} \, dx=\frac{\pi t-4 \log (t+1)+\log (4)}{4 t^2+4}$$