Introducing Integral Calculus
Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!
Calculus - Introducing Integral Calculus - Outline of Contents:
Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions. Definite integration and properties of definite integrals; Integration by substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration.
Here's a quick look at the topics which have been introduced in this tutorial. This tutorial introduces, formulas and definitions - future tutorials will cover examples and problems based on these. The relevance of these rules and formulas will be better appreciated as you move on to solving problems based on these, in subsequent tutorials. Definitions and Formulas - Indefinite and Definite Integrals, Substitution, Integration by Parts, Reduction Formulas and Summation of Series
• Integral of a Function:
If the derivative of a function f (x) is F (x), i.e, if df (x)/dx = F (x), we say that f (x) is an Integral of F (x) and, in symbols, we write ∫F (x) dx = f (x).
The letter x in dx, denotes that the integration is to be performed with respect to the variable x. The process of determining an integral of a function is called Integration and the function to be integrated is
called Integrand . Integration and differentiation are inverse processes.
Integrals of common trigonometric and hyperbolic functions like sin x, cos x, tan x, sec x tan x, cosec x cot x, sinh x, cosh x Integrals of common inverse functions such as 1/(x2-a2)0.5, 1/(x2+a2)0.5 , 1/(x2+a2), 1/(a2-x2)0.5 etc.
• Indefinite Integral(General Integral):
If f (x) is an integral of F (x), then f (x) + c is the indefinite integral of the function F (x), where the constant c is called Constant of integration.
∫F (x) dx = f (x) + c.
Integral of a function is not unique and that if f (x) be any one integral of F (x), then
(i) f (x) + c is also its integral; c being any constant whatsoever.
(ii) every integral of F (x) can be obtained from, f (x) + c, by giving some suitable value to c.
Thus if f (x) be any one integral of F (x), then f (x) + c is its General integral .
• The tutorial will introduce some Elementary Integrals, Constant , Sum , Difference and Power rules; Trigonometric, Hyperbolic and Inverse functions
Special integrals of functions in the form of:
∫ f'(x)/f(x) dx ∫ [f(x)]nf'(x) dx ∫ ex [f(x) + f'(x)] dx ∫ eax sin(bx + c) dx ∫ eax cos(bx + c) dx
• Definite Integral:
In Geometric and other applications of Integral Calculus, it becomes necessary to find the difference in the values of an integral of a function f (x) for two assigned values of the independent variable x, say,a,b. This difference is called the Definite Integral of f (x) over the interval [a, b] and equals F (b) − F (a), where F (x) is an integral of f (x). The number, a, is called the Lower limit and the number, b, is called the Upper limit of integration. The value of a definite integral is unique and is independent of the particular integral which we may employ to calculate it.
This is also, the area under a curve, between x = a and b.
We will also introduce some of the formulas related to the properties of definite integrals.
• Integration by substitution:
This method consists in expressing the integral ∫f (x) dx, where x is the independent variable, in terms of another integral where some other variable, say t, is the independent variable; x and t being connected by some suitable relation x = φ(t).
• Important forms of integrals, Important Integrals, Definite Integral as the limit of a sum.
• Integration by parts:
If u = f (x), v = g(x), and the differentials du = f (x) dx and dv = g (x) dx, then integration by parts states that ∫ u dv = uv − ∫ v du
Where u and v are chosen as per the ILATE rule. LIATE Rule: u should be chosen such that which function comes first from the following list : L : Logarithmic functions, I : Inverse Trigonometric functions, A : Algebraic functions, T : Trigonometric functions, E : Exponential functions.
• Reduction Formulae:
A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem, and then reducing that to the problem of solving an easier problem, and so on. The reduction problem is generally obtained by applying the rule of integration by parts.
• Summation of series:
It is possible to express the limits of sums of certain types of series as definite integrals, and thus to evaluate them. It is always possible to so transform such a series that the lower and upper limits of the corresponding definite integral are 0 and 1 respectively.
Complete Tutorial with formulas, definitions and important lists :
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