Help to Integral Calculus Solutions & Examples

I need help computing this integral:
$\begin{array}{}\text{(1)}& {\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}\frac{x\cos <!--\; \; -->(ax)}{x^2+1}\coth <!--\; \; -->(sx)\mathrm{d}x\end{array}$

What is the distance between the following polar coordinates?: ${\displaystyle (1,\frac{3\pi }{4}),(5,\frac{5\pi }{8})}$

I was trying to solve:
$\begin{array}{}& {\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}\cos <!--\; \; -->(x^m)x^n\mathrm{d}x\end{array}$

Determine $M$ for Simpson's rule error bound: $2e^x+x^3$ where $a=0$ and $b=1.6$.

I've been told numerical methods in solving ODEs, such as Euler's method and Runge-Kutta, are all in some way approximations to Picard's iteration, and I'm trying to understand how.

Suppose we have a differential equation on an interval $[x_0,x_L]$:
$\frac{dy}{dx}=f(x,y)$
with initial condition $y(x_0)=y_0$

I would like to numerically solve the equation on a set of points $\{x_0<x_1<\cdots <!--\; \cdots \; --><x_n\}$, i.e. obtain approximations $y_i$ to the true solution $y(x_i)$ for each $x_i$.
Picard's iteration works as follows:
$y_{0,0}=y_0$
$y_{0,k}(x_1)=y_0+{\int <!--\; \int \; -->}_{x_0}^{x_1}f(x,y_{0,k-<!--\; -\; -->1}(x))dx\mathrm{f}\mathrm{o}\mathrm{r}k\ge <!--\; \ge \; -->1$
suppose we stop for $k=m$, then take $y_1=y_{0,m}(x_1)$
We then repeat the process for $i\ge <!--\; \ge \; -->1$.
$y_{i,0}=y_i$
$y_{i,k}(x_{i+1})=y_i+{\int <!--\; \int \; -->}_{x_i}^{x_{i+1}}f(x,y_{i,k-<!--\; -\; -->1}(x))dx\mathrm{f}\mathrm{o}\mathrm{r}k\ge <!--\; \ge \; -->1$
$y_{i+1}=y_{i,m}(x_{i+1})$
So is the idea of a numerical method (e.g. Euler's method) to replace the integral ${\int <!--\; \int \; -->}_{x_i}^{x_{i+1}}f(x,y_{i,k-<!--\; -\; -->1}(x))dx$ with an approximation such as $(x_{i+1}-<!--\; -\; -->x_i)f(x_i,y_i)$ (Euler's method)? What I don't understand is why numerical methods only iterate once for each point $x_i$ (in other words, $m=1$) but Picard's iteration suggests you should iterate multiple (potentially many) times for each $x_i$?

The problem is to prove that:
${\displaystyle {\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }(a\sin \left(x\right)+b\cos \left(x\right)+c\sin \left(y\right)+d\cos \left(y\right))e^{-2\cos (y-x)+a\cos \left(x\right)-b\sin \left(x\right)+c\cos \left(y\right)-d\sin \left(y\right)}dxdy}$
I have an idea to represent a function under the integral like the odd function, but I can't.
Also, I can prove, that: ${\displaystyle {\int }_{-\pi }^{\pi }(a\cos \left(x\right)+b\sin \left(x\right))e^{a\sin \left(x\right)-b\cos \left(x\right)}dx=0}$

Find the integral: $\int <!--\; \int \; -->-<!--\; -\; -->\sqrt{t^2+t-<!--\; -\; -->5}dt$.

Evaluate $\int <!--\; \int \; -->(\frac{1}{x^4}-<!--\; -\; -->\frac{1}{2x^2}-<!--\; -\; -->\frac{5x}{2}-<!--\; -\; -->3\sin <!--\; \; -->(2x))dx$.

Integrate $\frac{2\sqrt{{\sin }^{-<!--\; -\; -->1}<!--\; \; -->(6t)}}{3\sqrt{1-<!--\; -\; -->36t^2}}$ with respect to t.

Integrate $-<!--\; -\; -->\frac{e^t}{\sqrt{4e^{2t}-<!--\; -\; -->1}}$ with respect to t.

Find the integral: $\int <!--\; \int \; -->\frac{1}{\sqrt{x^2+x+1}}dx$.

Evaluate $\int <!--\; \int \; -->\frac{2\cos <!--\; \; -->(2t)}{3(\sin <!--\; \; -->(2t)-<!--\; -\; -->2{)}^9}dt$.

Find the integral: $\int <!--\; \int \; -->\frac{1}{\sqrt{2t^2-<!--\; -\; -->6t+1}}dt$.

Evaluate $\int <!--\; \int \; -->-<!--\; -\; -->\frac{1}{\sqrt{2x^2-<!--\; -\; -->2x}}dx$.

Evaluate $\int <!--\; \int \; -->-<!--\; -\; -->2e^t\sqrt{9-<!--\; -\; -->e^{2t}}dt$.

Find the integral: $\int <!--\; \int \; -->\frac{2e^x}{\sqrt{e^{2x}+4}}dx$.

Compute $\int <!--\; \int \; -->e^t\sqrt{9e^{2t}+4}dt$.

Integrate $\frac{4e^{{\tan }^{-<!--\; -\; -->1}<!--\; \; -->(t)}}{7(t^2+1)}$ with respect to t.

Integrate $\frac{8}{7}{x}^3\tan <!--\; \; -->(x^4+4)\sec <!--\; \; -->(x^4+4)$ with respect to x.

Evaluate $\int <!--\; \int \; -->\frac{5{\tan }^{-<!--\; -\; -->1}<!--\; \; -->(4x{)}^{10}}{3(16x^2+1)}dx$.