Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

Rotation matrix to construct canonical form of a conic
$C : 9 x^{2} + 4 x y + 6 y^{2} - 10 = 0 .$
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial
$p (t) = \det (\begin{matrix} 9 - t & 2 \\ 2 & 6 - t \end{matrix})$
The eigenvalue are $t_{1} = 5, t_{2} = 10$ , with associated eigenvectors $(- 1, 2), (2, 1)$ . Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that $det (R) = 1$ ):
$R = \frac{1}{\sqrt{5}} (\begin{matrix} 1 & 2 \\ - 2 & 1 \end{matrix})$
Then ${(x, y)}^{t} = R {(x^{'}, y^{'})}^{t}$ , and after some computations I find the canonical form
$\frac{1}{2} x^{' 2} + \frac{4}{5} y^{' 2} = 1 .$

Averie Ferguson 2022-03-31

Solving periodic equation $\cos 7 θ = \cos 3 θ + \sin 5 θ$
I approached the problem as follows : $\cos 7 θ = \cos 3 θ + \sin 5 θ \Rightarrow \cos 7 θ - \cos 3 θ = \sin 5 θ \Rightarrow 2 \sin 5 θ \sin (- 2 θ) = \sin 5 θ$
From there on we get, $\sin 2 θ = - \frac{1}{2} and \sin 5 θ = 0$
So, we deduce that $θ = - \frac{π}{12} + π k$ and $θ = \frac{2}{5} π n$ where k and n are integers.
However, Wolfram|Alpha doesn't seem to agree with me. They present $\frac{π}{5}$ as a solution which is not included in my solution.

Lucian Ayers 2022-03-31

$\int \frac{d x}{1 + x}, x \geq 0$ by trigonometric substitution
Consider the integral $I = \int \frac{d x}{1 + x}, x \in [0, \infty)$ . A standard treatment using the substitution u=1+x directly gives the result $\ln (1 + x) + c$ . Now consider doing this with the trigonometric substitution $\sqrt{x} = \tan θ, θ \in [0, \frac{π}{2})$ then $x = \tan^{2} θ, d x = 2 \tan θ \sec^{2} θ$ . Now, following your nose with right triangle trigonometry this leads directly to the solution $I = 2 \ln (1 + x) + c$ .

svezic0rn 2022-03-31

Proving the double differential of $z = - z$ implies $z = \sin x$
$\frac{d^{2} z}{{d x}^{2}} = - z$
implies z is of the form $a \sin x + b \cos x$ . Is there a proof for the same. I was trying to arrive at the desired function but couldn't understand how to get these trigonometric functions in the equations by integration. Does it require the use of taylor polynomial expansion of $\sin x or \cos x$ ?

Hugh Soto 2022-03-31

Shape induced by the condition of angle bisector passing through a fixed point
Let A, B be two points on 2D plane. For any $C \in \overset{―}{A B}$ , define the set $S = {P; ∣; ∠ A P C = ∠ C P B} .$
What is the shape of S?

Polchinio3es 2022-03-30

$(2 x + \sqrt{4 x^{2} + 1}) (\sqrt{y^{2} + 4} - 2) \geq y > 0$ minimum vale of $x + y$

dirafe7455 2022-03-29

A particle moves along a line with velocity function v(t)=t^2-t, where v is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0,5].

Oxinailelpels3t14 2022-03-29

How to solve system of equations $A_{1} x^{2} + B_{1} x y + C_{1} y^{2} + D_{1} x + E_{1} y + F_{1} = 0$ and $A_{2} x^{2} + B_{2} x y + C_{2} y^{2} + D_{2} x + E_{2} y + F_{2} = 0$
How to express x through y?

blestimd4pz 2022-03-29

How to solve $\int \sin^{3} (x) \cos^{2} (x) d x$ with integration by parts?

Find a vector function that represents the curve of intersection of the two surfaces of the cylinder x^2+y^2=4 and the surface z=xy.

Aishy Chakraborty2022-03-28

Pizzadililehz 2022-03-27

Coordinate geometry and Trignometry.
Find the condition so that the line $p x + q y = r$ intersects the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in points whose eccentric angles differ by $\frac{π}{4}$ .
Though I know how to solve it using parametric coordinates, I was wondering if there's an another approach which is less time consuming.

ikramkeyslo4s 2022-03-27

How Can you factor 1/2 out of integral of $\cos 2 x$ ?
$\int \cos 2 x d x$
simplifies to
$\frac{1}{2} \int \cos u d u$
where u=2x
My Question is where does the $\frac{1}{2}$ that is factored out come from?
It would make more sense for a factor of 1/2 to be extracted if the integral was:
$\int \frac{\cos x}{2} d x$

Coradossi7xod 2022-03-27

Given: $y = x^{1.65}$
Could it be described as parabolic in shape, or does the equation have to have $x^{2}$ as its highest degree term?

ropowiec2gkc 2022-03-26

Determine whether $(x, y) = (m \cos (ω t), n \cos (ω t - ϕ))$ describes an ellipse for all m, n, $ω, ϕ$

rijstmeel7d4t 2022-03-26

How can you prove that the midpoint of the square will be Origin?
Vertices A,B,C,D and the midpoints of the sides of the square ABCD lie on $x^{2} y^{2} = 1$ .
No other points of the square lies on the curve.
How can you prove that the midpoint of the square will be Origin ?
My approach: If I can show if (x,y) are the coordinates of A then (y, -x) will be so of B , I will be done. I supposed A lies on first quadrant , B lies on second quadrant etc..
Can anyone tell please me how to show it ?

svezic0rn 2022-03-25

Definite Integral $\int_{\frac{π}{12}}^{\frac{π}{4}} \frac{8 \cos 2 x}{{(\tan x + \cot x)}^{3}} d x$

ideklaraz7xz 2022-03-25

Why does $\int_{0}^{n π} \frac{d x}{1 + \tan^{2 k} (x)} = n \frac{π}{2}$ hold for all non-negative integer k?

zatajuxoqj 2022-03-24

Formula for analytical finding ellipse and circle intersection points if exist
I need a formula that will give me all points of random ellipse and circle intersection (ok, not fully random, the center of circle is laying on ellipse curve)
I need step by step solution (algorithm how to find it) if this is possible.

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