PSI Mathematics.

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).

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