Integration Chapter Notes Class 12 Cbse Mathematics

Chapter: IntegrationIntegral Calculus1 IntroductionIntegration is one of the two fundamental operations of calculus, the other being differ-entiation. While differentiation deals with the rate of change of a quantity, integrationdeals with accumulation. Historically, integration arose from problems of finding areas,volumes, arc lengths, and physical quantities such as work and mass. Mathematically, integration is the inverse process of differentiation.2 Indefinite Integral2.1 DefinitionLetf(x) be a real-valued function defined on an intervalI. If there exists a functionF(x) such that d dxF(x) =f(x),thenF(x) is called anantiderivativeoff(x). The collection of all antiderivatives off(x) is called theindefinite integraland isdenoted by Zf(x)dx=F(x) +C,whereCis an arbitrary constant.2.2 Basic Properties 1. Linearity: Z(af(x) +bg(x))dx=a Zf(x)dx+b Zg(x)dx2. Constant of integration must always be included.2.3 Solved ExampleExample:Evaluate R5x 4dx.Solution: Z5x4dx= 5 x5 5+C=x5+C13 Standard IntegralsSome important standard integrals are listed below: Zxndx= xn+1 n+ 1+C, n=1Z 1 xdx= ln|x|+CZ exdx=e x+CZ axdx= ax lna+C, a >0, a= 1Z sinx dx=cosx+CZ cosx dx= sinx+CZ sec 2x dx= tanx+C4 Methods of Integration4.1 Integration by SubstitutionThis method is based on reversing the chain rule.4.1.1 FormulaIf Zf(g(x))g (x)dx,then putu=g(x), so thatdu=g (x)dxandZf(u)du4.1.2 ExampleEvaluate R2xcos(x 2)dx.Solution:Letu=x 2, thendu= 2x dx.2Z2xcos x 2dx= Zcosu du= sinu+C= sin x 2+C4.2 Integration by Parts4.2.1 FormulaDerived from the product rule: Zu