PSI Mathematics.

What is 0/0 ?

Offcourse, it is undefined by mathematical analogy.

Now ,what if following assertarions take place.

What if tan90⁰/tan90⁰ = 1 or 0/0.

What if 0⁰c/0⁰c = 1 or 0/0.

Sin0⁰/Sin0⁰ = 0/0 or 1

Cos90⁰/Cos90⁰ = 0/0, 1

Similary, there is number of such possiblities.

Fibabaoci Sequence

Integration by Partial Fractions Formula
To find the integral of an improper fraction like P(x)/Q(x), in which the degree of P(x) < that of Q(x), we can use integration by partial fractions. In this method, we split the fraction using partial fraction decomposition as P(x)/Q(x) = T(x) + P11 (x)/ Q(x), in which T(x) is a polynomial in x and P11 (x)/ Q(x) is a proper rational function. Assume that A, B and C are real numbers, we can have the following types of simpler partial fractions associated with various types of rational functions.
Rational Fractions Partial Fractions

(px + q)/(x-a)(x – b) A/(x – a) + B/ (x-b)
(px + q)/(x-a)n A1/(x-a) + A2/(x-a)2 + ………. An/(x-a)n
(px2 + qx + r)/(ax2 + bx + c)n (A1x + B1)/(ax2 + bx + c) + (A2x + B2)/(ax2 + bx + c)2 + …(Anx + Bn)/(ax2 + bx + c)n
(px2 + qx + r)/(ax2 + bx + c) (Ax + B)/(ax2 + bx + c)
(px2 + qx + r)/(x-a)(x-b)(x-c) A/(x – a) + B/ (x-b) + C/ (x-c)
(px2 + qx + r)/(x2 +bx +c) A/(x-a) +(Bx+C)/(x2 +bx +c).

Integration Formulas of Inverse Trigonometric functions:
∫1/√(1 – x2).dx = sin-1x + C
∫ /1(1 – x2).dx = -cos-1x + C
∫1/(1 + x2).dx = tan-1x + C
∫ 1/(1 +x2 ).dx = -cot-1x + C
∫ 1/x√(x2 – 1).dx = -cosec-1 x + C
∫ 1/x√(x2 – 1).dx = sec-1x + C
Advanced Integration Formulas
∫ 1/(a2 – x2).dx =1/2a.log|(a + x)(a – x)| + C
∫1/(x2 – a2).dx = 1/2a.log|(x – a)(x + a| + C
∫1/(x2 + a2).dx = 1/a.tan-1x/a + C
∫1/√(x2 – a2)dx = log|x +√(x2 – a2)| + C
∫1/√(a2 – x2).dx = sin-1 x/a + C
∫ √(x2 – a2).dx =1/2.x.√(x2 – a2)-a2/2 log|x + √(x2 – a2)| + C
∫√(a2 – x2).dx = 1/2.x.√(a2 – x2).dx + a2/2.sin-1 x/a + C
∫1/√(x2 + a2 ).dx = log|x + √(x2 + a2)| + C
∫ √(x2 + a2 ).dx =1/2.x.√(x2 + a2 )+ a2/2 . log|x + √(x2 + a2 )| + C
Different Integration Formulas
T3 types of integration methods are generally used: Integration by parts formula, Integration by Substitution formula and Integration by partial fractions formula. Let us look at each of these integration formulas one by one.

Integration by Parts Formula
When any given function is a product of two different functions, the integration by parts formula or partial integration can be applied to evaluate the integral. The integration formula using partial integration methos is as follows:

∫ f(x).g(x) = f(x).∫g(x).dx -∫(∫g(x).dx.f'(x)).dx + c

For instance: ∫ xex dx is of the form ∫ f(x).g(x). Therefore, we must apply the appropriate integration formula and evaluate the integral accordingly.

f(x) = x and g(x) = ex

Thus ∫ xex dx = x∫ex .dx – ∫( ∫ex .dx. x). dx+ c

= xex – ex + c

Integration by Substitution Formula
If a given function is a function of another function, we can apply the integration formula for substitution to solve that integral. For instance, if
I = ∫ f(x) dx,
where
x = g(t) so that dx/dt = g'(t), then we write dx = g'(t)
Take for instance
I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt
For example: Consider ∫ (3x +2)4 dx
The integration formula of substitution is given as follows.
Take u = (3x+2). ⇒ du = 3. dx
Thus ∫ (3x +2)4 dx =1/3. ∫(u)4. du
= 1/3. u5 /5 = u5 /15
= (3x+2)5 /15

Basic Integration Formulas
Some generalized results obtained using the fundamental theorems of integrals are remembered as integration formulas in indefinite integration.

∫ xn.dx = x(n + 1)/(n + 1)+ C
∫ 1.dx = x + C
∫1/x.dx = log|x| + C
∫ ex.dx = ex + C
∫ ax.dx = ax /loga+ C
∫ ex[f(x) + f'(x)].dx = ex.f(x) + C
Integration Formulas of Trigonometric functions
We simplify and rewrite trigonometric functions as functions that are integrable. A list of trigonometric functions in integration is given below:

∫ cosx.dx = sinx + C
∫ sinx.dx = -cosx + C
∫ cosec2x.dx = -cotx + C
∫ sec2x.dx = tanx + C
∫ cosecx.cotx.dx = -cosecx + C
∫ secx.tanx.dx = secx + C
∫ tanx.dx =log|secx| + C
∫ cotx.dx = log|sinx| + C
∫ cosecx.dx = log|cosecx – cotx| + C
∫ secx.dx = log|secx + tanx| + C

GEOMETRICAL AXIOM
1 : Things which are equal to same things equals to one another.

e:g , if perimeter of triangle is equal to perimeter rectangle and perimeter rectangle is equal that of square then the perimeter of triangle is equal to perimeter of square .

2 : if equals are added to equals , then the whole are equal .
E:g , magnitude of same kind can be added and compared, a = b , then a+c = b+c .

3 : if equals are subtracted from equals then reminders are equal.
E:g , 2x = 2y implies 2x-3 = 2y-3 .

4 : Things which are coincide with one another are equal to one another.

E:g , if two parallel lines tends to each other limit of distance reduce having convergence , its coincide with each other .

5 : the whole is greater than then the part .

E:g , the right angles greater than the acute angles , that of straight angle is greater than the right angle similarly, reflexive angle is greater than straight angle and that of complete angle is greater than reflexive angle.

6 : things which are double of something are equal to one another .

E:g , if y = 2x , z= 2x , then y = z .

7 : things which are halves of the same things are equal to one another .

E:g , right angles which are halves of straight angles are equal to one another .

8 : if equals are multiplied by equals then their products are equal .

E:g , if x = y then 5x = 5y .

9 : if equals are divided by equals , then their quotients are equal.

E:g , if x = y , then x/a = y/a .

10 : of two quantities of same kind , the first is greater then the second, this axiom is called "trichotomy law" .

E:g , x > y , x = y , x

BEING

Being, pure being - without further determination. In its indeterminate immediacy it is equal only to itself and also not unequal with respect to another; it has no difference within it, nor any outwardly. If any determination or content were posited in it as distinct, or if it were posited by this determination or content as distinct from an other, it would thereby fail to hold fast to its purity. It is pure indeterminateness and emptiness. - There is nothing to be intuited in it, if one can speak here of intuiting; or, it is only this pure empty intuiting itself. Just as little is anything to be thought in it, or, it is equally only this empty thinking. Being, the indeterminate immediate is in fact nothing, and neither more nor less than nothing.

NOTHING

Nothing, pure nothingness; it is simple equality with itself, complete emptiness, complete absence of determination and content; lack of all distinction within. - In so far as mention can be made here of intuiting and thinking, it makes a difference whether something or nothing is being intuited or thought. To intuit or to think nothing has therefore a meaning; the two are distinguished and so nothing is (concretely exists) in our intuiting or thinking; or rather it is the empty intuiting and thinking itself, like pure being. - Nothing is therefore the same determination or rather absence of determination, and thus altogether the same as what pure being is'.

BECOMING

Unity of being and nothing

Pure being and pure nothing are therefore the same. The truth is neither being nor nothing, but rather that being has passed over into nothing and nothing into being - 'has passed over', not passes over. But the truth is just as much that they are not without distinction; it is rather that they are not the same, that they are absolutely distinct yet equally unseparated and inseparable, and that each immediately vanishes in its opposite. Their truth is therefore this movement of the immediate vanishing of the one into the other: becoming, a movement in which the two are distinguished, but by a distinction which has just as immediately dissolved itself'.

No cat has eight tails.
One cat has one tail .

Adding = one cat has nine tail .

Which kind of fallacy is in the above statement ?

Number of Indian people who drenched in rain doesn't equal to number of Indian people who dranched in rain 🌧️ ...

Is the stament logically correct ?

Only mathematicians can make the balance between mistress and wife ...😁

Why does be so ?