Help to Integral Calculus Solutions & Examples

Integral of ${\displaystyle {\sin }^5\left(x\right)\cos \left(x\right)}$
I assumed I would do u-substitution where:
${\displaystyle u=\sin \left(x\right)}$
${\displaystyle du=\cos \left(x\right)dx}$
Which would then cancel out the ${\displaystyle \cos \left(x\right)}$
And leave me with:
${\displaystyle \int u^{5}du=\frac{u^6}{6}+C=\frac{{\sin }^6\left(x\right)}{6}+C}$

Integrating ${\displaystyle \int \frac{1}{\cos \left(\theta \right)}d\theta }$

Integrate ${\displaystyle {\int }_0^{\pi }\sqrt{\frac{1+\cos \left(2x\right)}{2}}dx}$
It is obvious that by using trig identities we can come up with:
${\displaystyle {\int }_0^{\pi }\sqrt{{\cos }^2\left(x\right)}=}$
${\displaystyle ={\int }_0^{\pi }\left|\cos \left(x\right)\right|dx}$

Integrate ${\displaystyle \int \frac{dx}{1+\sin x}}$
My attempt:
${\displaystyle =\int \frac{dx}{1+\sin x}}$
${\displaystyle =\int \left\{\frac{dx}{1+\frac{2\tan \left(\frac{x}{2}\right)}{1+{\tan }^2\left(\frac{x}{2}\right)}}\right\}}$
${\displaystyle =\int \left\{\frac{1+{\tan }^2\left(\frac{x}{2}\right)}{{(1+\tan \left(\frac{x}{2}\right))}^2}\right\}dx}$

How to find the position on ellipse (or hyperbola) arc if we know it's euclidean distance from given point and direction of movement?

How to find the axes of a rotated ellipse
I have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x' and y') and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?
I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I've only managed this through very many tedious iterations and it isn't exact -- is there a better way? Thank you.

How to compute the integral of ${\displaystyle \int \sin (1-x^2)dx}$
I tried substitution, ${\displaystyle u=1-x^2,du=-2xdx,dx=-\frac{du}{2x}}$, yielding:
${\displaystyle \int \sin u(-\frac{du}{2})}$
${\displaystyle =-\frac{1}{2}\int \sin \left(u\right)du}$
${\displaystyle =-\frac{1}{2}\cos u}$
${\displaystyle =-\frac{1}{2}\cos (1-x^2)}$
WolframAlpha is showing something completely different, however. What's wrong with my solution?

What is the complex form through five points of a conic section in the complex plane?

Solve ${\displaystyle {\int }_{\left|z\right|=3}\tan \left(\pi z\right)dz}$ using argument principle
${\displaystyle {\int }_{\left|z\right|=3}\tan \left(\pi z\right)dz=\int \frac{d\left(\cos \pi z\right)}{(-\pi )\cos \pi z}dz=-2i(N-P)}$
where N = num of zeroes inside C:|z| = 3 and P is num of poles inside C (Is this correct or should we also consider on C???)
zeroes for ${\displaystyle \cos \pi z=-0.5,-1.5,-2.5,0.5,1.5,2.5\Rightarrow N=6}$
No poles for ${\displaystyle \cos \pi z}$
value of given integral = ${\displaystyle -2i(N-P)=-2i(6-0)=-12i}$
Is this correct? pls correct me if i am doing wrong

What can you say about the motion of an object with velocity vector perpendicular to position vector? Can you say anything about it at all?
I know that velocity is always perpendicular to the position vector for circular motion and at the endpoints of elliptical motion. Is there a general statement that can be made about the object's motion when the velocity is perpendicular to position?

What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?
How will look the general conic sections equation ${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola?
I need a formulas for coefficients A, B, C, D, E and F for ellipse, hyperbola and parabola. I did it for not-rotated conic sections at the origin of coordinates but have a difficults with rotated and shifted.
For i.e. standart ellipse equation ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ gives me general equation ${\displaystyle \frac{1}{a^2}{x}^2+0xy+\frac{1}{b^2}{y}^2+0x+0y-1=0}$
I need the same in case if ellipse located in position (h;k) and rotated on some angle ${\displaystyle \alpha }$ from positive X axis. And the same for parabola and hyperbola.

Is ${\displaystyle x^2=4ay}$ a function while ${\displaystyle y^2=4ax}$ is not?

Is ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\int }_0^{\frac{\pi }{2}}\frac{x^{\frac{1}{n}}+\sin x}{\tan x+x^{\frac{1}{n}}}dx={\int }_0^{\frac{\pi }{2}}\frac{1+\sin x}{\tan x+1}dx=1.62..}$?
Note: The LHS of this identity it's seems converges for every positive integer ${\displaystyle n\ge 1}$ over the range ${\displaystyle (0,\frac{\pi }{2})}$

Find semi-major/semi-minor axis of ellipse from parametric equations with different phase
${\displaystyle x=\hat{x}\cdot \cos (\Omega t-\theta )}$
${\displaystyle y=\hat{y}\cdot \sin (\Omega t-\phi )}$

Is there any (elementary) way to evaluate the integral
${\displaystyle {\int }_0^{\frac{\pi }{2}}{\left(\cos x\right)}^a{(1-\cos x)}^{1-a}dx}$
for any ${\displaystyle a\in (0,1)}$?

Is there a parametrization of a hyperbola ${\displaystyle x^2-y^2=1}$ by functions x(t) and y(t) both birational?
Consider the hyperbola ${\displaystyle x^2-y^2=1}$. I am aware of some parametrizations like:
1. ${\displaystyle (x\left(t\right),y\left(t\right))=(\frac{t^2+1}{2t},\frac{t^2-1}{2t})}$
2. ${\displaystyle (x\left(t\right),y\left(t\right))=(\frac{t^2+1}{t^2-1},\frac{2t}{t^2-1})}$
3. ${\displaystyle (x\left(t\right),y\left(t\right))=(\text{cosh}t,\text{sinh}t)}$
4. ${\displaystyle (x\left(t\right),y\left(t\right))=(\sec \left(t\right),\tan \left(t\right))}$
The first and the second are by rational functions x(t) and y(t). But the functions are not birational(i.e. with rational inverse of each).
Is there a parametrization where:
- x(t) is rational with inverse also rational, and
- y(t) is rational with inverse also rational?
Is possible, to find a parametrization where both are rational and at least one of the has inverse rational?

How do I integrate the following:
${\displaystyle \int \frac{\sqrt{\tan ax}}{1+\sqrt{\tan ax}}dx}$

Find the parametric equation for the tangent line to the intersection curve between an ellipsoid and a paraboloid?
We got various problems in this site asking for similar problems btw an ellipsoid and a plane. What if it's btw an ellipsoid and a paraboloid?
I got the equation of both surfaces: ${\displaystyle 4x^2+y^2+z^2=9}$ and ${\displaystyle z=x^2+y^2}$ respectively. Now given the tangent line touches their intersection (as an off-origin xz-ellipse but this might be useless) at (-1,1,2), how can we find the v- for the parametric equation of the tangent line ${\displaystyle \underset{―}{r}\left(t\right)=(-1,1,2)+t\underset{―}{v}}$? Is there any way we need to utilize the directional derivative of either surface? Or we just use f(x,y,z) and g(x,y,z) to describe both surfaces so that ${\displaystyle \underset{―}{v}=\mathrm{\nabla }f\times \mathrm{\nabla }g}$ which I got after simplification as (5,8,6)? Thanks in advance.

Find domain and range of the slanted hyperbola
Given the conic section ${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$ and I know that it is a hyperbola and ${\displaystyle B\ne 0}$.
How to find its domain and range? I guess the method of Lagrange multipliers will fail here.