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What is meant by integration in mathematics?

Integration, in mathematics, technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign “∫,” as in ∫f(x), usually called the indefinite integral of the function. The symbol dx represents an infinitesimal displacement along x; thus ∫f(x)dx is the summation of the product of f(x) and dx. The definite integral, written

with a and b called the limits of integration, is equal to g(b) − g(a), where Dg(x) = f(x).

Some antiderivatives can be calculated by merely recalling which function has a given derivative, but the techniques of integration mostly involve classifying the functions according to which types of manipulations will change the function into a form the antiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/(x + 1) can be easily recognized as the derivative of loge(x + 1). The antiderivative of (x2 + x + 1)/(x + 1) cannot be so easily recognized, but if written as x(x + 1)/(x + 1) + 1/(x + 1) = x + 1/(x + 1), it then can be recognized as the derivative of x2/2 + loge(x + 1). One useful aid for integration is the theorem known as integration by parts. In symbols, the rule is ∫fDg = fg − ∫gDf. That is, if a function is the product of two other functions, f and one that can be recognized as the derivative of some function g, then the original problem can be solved if one can integrate the product gDf. For example, if f = x, and Dg = cos x, then ∫x·cos x = x·sin x − ∫sin x = x·sin x − cos x + C. Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

Section 1Introduction
Lecture 1Introduction
Lecture 2Introduction Continued
Section 2Integration as process of differentiation
Section 3Geometrical interpretion indefinite integrals
Section 4 MatProperties of indefinite integrals
Section 5 MathsComparision of differentiation & Integration
Section 6 Maths InIndefinite integrals by inspection
Lecture 7 Maths InIndefinite integrals by inspection
Lecture 8Example:Indefinite integrals by inspection
Lecture 9Example:Indefinite integrals by inspection
Section 7Integration by substitution
Lecture 10Integration by substitution
Lecture 11Example: Integration by substitution
Lecture 12Example: Integration by substitution
Section 8Integration by substitution: Formula
Lecture 13Integration by substitution: Formula
Lecture 14 Maths IExample: Integration by substitution
Lecture 15 Maths IExample: Integration by substitution
Lecture 16 Maths IIntegration by substitution: Formula
Lecture 17Integration by substitution 1
Lecture 18Integration by substitution 2
Lecture 19Integration by substitution 3
Lecture 20Integration by substitution 4
Section 9 MathIntegration by trigonometric identities
Lecture 21 MathIntegration by trigonometric identities
Lecture 22 MExample Integration trigonometric Identities
Lecture 23 MExample Integration trigonometric Identities
Section 10 MathIntegrals of some particular functions
Lecture 24 MathIntegrals of some particular functions
Lecture 25Proof:Integrals of some particular
Lecture 26Proof:Integrals of some particular functions
Section 11Integration of special types
Lecture 27Integration of special types
Lecture 28 Maths IntegralExample Integrals some particular functions
Lecture 29 Maths IntegralExample Integrals some particular functions
Section 12Integration by partial fractions
Lecture 30Integration by partial fractions
Lecture 31Example:Integration by partial fractions
Lecture 32Example:Integration by partial fractions
Lecture 33Example:Integration by partial fractions
Section 13Integration by parts
Lecture 34Integration by parts
Lecture 35Example: Integration by parts
Lecture 36Example: Integration by parts
Lecture 37Example: Integration by parts
Section 14Definite integrals as limit of sum
Lecture 38Definite integrals as limit of sum
Lecture 39Example:Definite integrals as limit of sum
Lecture 40Example:Definite integrals as limit of sum
Section 15 MaFundamental theorem of integral calculus
Section 16Integral as area function
Section 17Definite integral by substitution)
Lecture 43Definite integral by substitution)
Lecture 44Example: Definite integral by substitution)
Section 18Properties of Definite integrals
Lecture 45Properties of Definite integrals
Lecture 46 MaProof: Properties of definite integrals
Lecture 47Solve problems using properties of definite integrals
Lecture 48Solve problems using properties of definite integrals