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Trigonometric Integration ${\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}\sin <!--\; \; -->(2x)\cos <!--\; \; -->(3x)dx$I think you need to use a double angle formula but I am not sure how I am supposed to split the 3x.

Prove that $\sin <!--\; \; -->x+\sin <!--\; \; -->y=1$ does not have integer solutions
Suppose x and y are angles measured in radians. Then how to show that the equation
$\sin <!--\; \; -->x+\sin <!--\; \; -->y=1$
does not have a solution $(x,y)\in <!--\; \in \; -->\mathbb{N}\times <!--\; \times \; -->\mathbb{N}$?
This question is prompted by curiosity. I don't have any ideas how it can be approached.

What is the average value of the function $f(x)=(x-<!--\; -\; -->1{)}^2$ on the interval from x=1 to x=5?

The integral of $\left|\frac{\cos <!--\; \; -->x}{x}\right|$
I'm looking to determine whether the following function is unbounded or not:
$F(x)={\int <!--\; \int \; -->}_1^x\left|\frac{\cos <!--\; \; -->t}{t}\right|\text{d}t$
I can't seem to do much with it because of the $|\cos <!--\; \; -->(t)|$. I thought of using the fact that $\int <!--\; \int \; -->|f|\ge <!--\; \ge \; -->|\int <!--\; \int \; -->f|$, but the problem is that the integral of $\frac{\cos <!--\; \; -->t}{t}$ (without the absolute values) is bounded, and so that doesn't prove that F(x) is unbounded or bounded. I tried re-expressing this as a cosine integral (the function Ci(x)) but to no avail. I'm not sure where else to go with this; the main problem seems to be the fact that its very difficult to derive an inequality with the $|\cos <!--\; \; -->(t)|$ without a $|\cos <!--\; \; -->(t)|$ on the other side of the inequality (or at least some trig function).

How to you find the general solution of $y{y}^{\prime }=\sin <!--\; \; -->x$?

What is the arc length of $f(x)=x^2-<!--\; -\; -->3x+\sqrt{x}$ on $x\in <!--\; \in \; -->[1,4]$?

What is the arc length of $f(x)=\frac{3x}{\sqrt{x-<!--\; -\; -->1}}$ on $x\in <!--\; \in \; -->[2,6]$?

How do you find the differential dy of the function $y=x\sqrt{1-<!--\; -\; -->x^2}$?

What is the integral of the momentum function?

What is the arc length of $f(x)=\frac{x^3}{6}+\frac{1}{2x}$ on $x\in <!--\; \in \; -->[1,3]$?

How do you find the volume of the region left of $y=\sqrt{2x}$ and below y=2 rotated about the y-axis?

How do you find the average value of $\sin <!--\; \; -->x$ as x varies between $[0,\mathrm{\pi <!--\; \pi \; -->}]$?

What is the surface area produced by rotating $f(x)=x^3-<!--\; -\; -->8,x\in <!--\; \in \; -->[0,2]$ around the x-axis?

Consider the function $f(x)=\frac{1}{\sqrt{x+2}}$ on the interval [-1, 23], how do you find the average or mean slope of the function on this interval?

Determine this real integral with the Residue-theorem.
${\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}\frac{\sin <!--\; \; -->x}{x^4-<!--\; -\; -->6x^2+10}\mathrm{d}x$

What is the surface area of the solid created by revolving
$f(x)=2x^2+3,x\in <!--\; \in \; -->[1,4]$ around the x axis?

What is the arc length of $f(x)=(x^3+x{)}^5$ in the interval [2,3]?

What is the arc length of $f(x)=\frac{x^2}{12}+x^{-<!--\; -\; -->1}$ on $x\in <!--\; \in \; -->[2,3]$?

How do you find the point c in the interval $0\le <!--\; \le \; -->x\le <!--\; \le \; -->2$ such that f(c) is equation to the average value of $f(x)=\sqrt{2x}$?

If R is the area enclosed by f(x) and g(x), is the volume of the solid generated by revolving R around the x-axis then revolving that solid around the y-axis equal to the volume of the solid generated if the order of the revolutions was switched?