As you said, you're in high-school. So, the most obvious method is to use the substitution method and the make an observation that the resulting integral involves the Exponential Integral of complex ...

Leibnitz's theorem basically employs the binomial theorem with powers representing orders of derivatives. $\dfrac{d^n}{dx^n}\left(x^n\ln(x)\right)=\sum_{k=0}^{n}\binom{n}{k}(x^n)^{(n-k)}\ln^{(k)}(x).$ ...

Find the marginal pmf of $m_i.$ Next, use the convolution theorem to find the pmf of $m_1+m_2,$ say, and see if this can be generalized to more than 2 random variables. Start and if you get stuck......

There is no way to represent $\int\cos(x^2)dx$ in terms of elementary functions! This is called the Fresnel integral, see Fresnel Integral. The best you can do is use series expansions.

Mathematica says the integral converges to $(2\sqrt{2}\pi^2)i$. So it probably involves a complex substitution. I'm working on it. I will let you know if I figure it out.
$\mathbb{I}(\cdot)$ is an indicator function I presume. In that case, your distribution does not have two parameters! It is only $\theta>0.$ Your distribution can actually be written in the ...