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# Mathematics : JEE Advanced Integration (2018-2019)

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JEE Advanced Integration (Topic and Subtopic)

1. Elementary integration, 2. advanced Integration by parts, 3. Integration by substitution, 4. advanced Integration using partial fractions, 5. Integrals containing quadratic (or higher order) equation in denominator, 6. Complex advanced integration, 7. advanced Integration of under-root of linear and quadratic functions, 8. Integr.. of exponential functions, 9. Trigonometric substitutions, 10. Reduction formula, 11. Definite integral as limit of sum, 12. Definite integral of odd and even functions, 13. Definite integral of periodic functions, 14. Step function, 15. Module function, 16. Integ.. using Leibnitz rule, 17. Using inequalities to solve definite integral, 18. Changing variable, 19. Interchanging and splitting limits, 20. 2a and a+b-x property, 21. Properties of definite integral, 22. Area under the curves, 23. Trapezoid rule and Simpson’s rule, 24. Order and degree of differential equation, 25. Formation of differential equation, 26. Homogeneous differential equation, 27. Linear and Bernoulli’s differential equation

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Formulas

Defi.. and Indefi.. Integral :  Indefi. integral of a function f(x) is a F(x) function whose derivative is f(x). Indefinite integral of a function is the primitive of the function. The terms indefinite integral, integral, primitive, and anti-derivative all mean the same thing. They are used interchangeably. Of the four terms, the term most commonly used is integral, short for indefinite integral. If F(x) is an integral of f(x) then F(x) + C is also an integral of f(x), where C is any constant.

Defini. Integrate f(x) : Integrate a function is to go through the process of searching the integral or primitive of the f(x).

Denote integrals : The notation used to denote the integral (or primitive) of a function f(x) is

$\int{f\left( x \right)\,dx}$

Why we use such a abstruse, confusing, cumbersome notation? Well, there are some reasons behind the notation.

If we assume we can use the usual algebraic rules in the manipulation of differentials in equation form. The notation follows as a logical consequence of the rules.

For Ex. consider the following.

The definition of an integral or primitive we have the following relationship between a function and its primitive F(x):

$f\left( x \right)=\frac{dF\left( x \right)}{dx}$

Then, using algebraic rules of manipulation,

d F(x) = f(x) dx

Taking the integral of both sides,

$\int{d\,f\left( x \right)}=\int{f\left( x \right)\,dx\,\,\,\,or\,\,\,\,\,F\left( x \right)}=\int{f\left( x \right)\,dx}$

Thus, using rules of algebraic manipulation, we have started with

$f\left( x \right)=\frac{dF\left( x \right)}{dx}$

and deduced that the primitive F(x) is given by

$F\left( x \right)=\int{f\left( x \right)\,dx}$

Thus this notation allows us to use algebraic manipulation in solving advanced integration problems.

In addition, most integration problems come in the form of definite integrals of the form

(I) $\int\limits_{a}^{b}{f\left( x \right)\,dx}$

and we work algebraically from that form. The source of the notation is undoubtedly the definite integral. It is the definite integral without the limits. Indeed, if you view the upper limit b of the definite integral 1) as variable, replace it with x, then it becomes the area function

(II) $A\left( x \right)=\int\limits_{a}^{x}{f\left( x \right)\,dx}$

and the area function A(x) does indeed represent a primitive of f(x).

Fundamental Integ.. formulas. The main formulas and rules for advanced integration, the most important of which need to be memorized. Many follow immediately from the standard differentiation formulas. a and m are constants.

Definition of derivative carried with it a formals process by which 1 could find derivative of a given function. Given whichever analytically clearly defined function (trigonometric, algebraic) we can search the derivative of it through a fairly straightforward process and the situation with advanced integration is different. There is no straightforward, direct process for integrating a function.

$\int{f\left( x \right)\,dx}$   ;  If we find a primitive F(x), then the integral is given as

$\int{f\left( x \right)}=F\left( x \right)+C$  ; where C is any constant.

Presented with a defi.. integral such as  $\int\limits_{a}^{b}{f\left( x \right)dx}$  evaluate, then the sol.. will have the form

$\left. \int\limits_{a}^{b}{f\left( x \right)dx=F\left( x \right)} \right|_{a}^{b}=F\left( b \right)-F\left( a \right)$

No guarantee that a solution exists that is expressible in terms of elementary functions. For Ex., none of the functions

$\sqrt{\sin x},\,\,\frac{1}{\ln x},\,\,\frac{\sin x}{x},\,\,{{e}^{-{{x}^{2}}}}$

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