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set pge tees lhe development of hitegral Calculus arises out of the effarts of sol ving the problems of the following types * The problem of finding a function whenever its derivative is piven. © The problem of finding the area bounded by the graph of a function under certain conditions. * ‘These two problems lead to the two forms of integrals i.e Indefinite Integrals and Definite Integrals Intepration is an Inverse Process of Differentiation: Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and ask to find its primitive i.e original function, Such a process is called integration or anti differentiation. nx+C]=cos x ty, d [sin x]=cos.x=> Joos xdx = sin x-+C => — dy dx dy, d wt Cl=x° and —|e*+C]=e" dy |x ] xan ck G +c] e * Anti derivatives or integrals of a function is not unique ¢ There exists infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. ¢ Cis customarily referred as arbitrary constant. ¢ In fact, Cis the parameter by varying which one gets different anti derivatives or integrals of a given function. e There is a function # such that f(r c}= se) vel x . © {F+C ,CeR} denote a family of anti derivatives of £ NOTATION - Given that a = S(x) = y= f fddx ax Syinbols/Terms/Phrases Meaning ff dx Integral of f with respect tox Sx) in Jr@ dx Integrand xin f£O) dx Variable of integration Integrate : Find the integral Ap integral of F A function F such that FM=asw) Integration, . The process of finding the integral Constant of Integration Any veal number C, considered as constant function GEOMETRICAL INTERPRETATION OF INTEGRAL CALCULUS- « Let f(x)=2x, then f2xdx= 2? +C, for different values of C, we get different integrals. These integrals are similar geometrically. » Thus, y=x’-+C(Cis an arbitrary constant), represents a family of integrals. By assigning different value of C’, we get different members of family. © yext-ly=x ype x41, y= x7 42 are the family of parabolas «© Ifthe line v= intersect all these parabolas, then slope of all these parabolas is 2a, this indicates that tangents fo all these curves are parallel. Scanned with CamScanner Surucys Aq wyoy saypour or > sport uolnnpnsqng 1 integral reduced to the form — == an a a da and hence can be evaluated, ve bate J pyrg Vo Mod the Mntegral af the type ax by-ve vonstanty, we me to find rea) numbers A, Ly such that dx, where p,q, a, b, ¢ are pyr rqon ‘ (av? bby be) +B A (ax 4b) + B dy ‘Yo determine A and 5, we equate fiom both sides the coefficients of x and the mt tans, A and Jae Us obtained and hence the integral is reduced te eon! one of the known foun . aX FO . | PENI ay ali to be done like j —f 1 dy type, Van? voere ax’ bebe INTEGRATION BY PARTIAL FRACTIONS- It is always possible to write the integrand as a sum of simpler tation functions by a inethod called partial function decomposition, Alter this integration can be carried out by using, suitable methods, ‘The tuble may be used for partial fractions. Scanned with CamScanner S.No.] Form of the rational function Torin of the paetial fraction 1. te P A_,u Gn-aiby 7? toa vob 2. Px a. A, —8_, Wwoay ena (wa) \ PR bayer Oa Web) wn FOX be where v7 by + ¢ cannot be factorised further INTEGRATION BY PARTS- It is called the product rule of integration. Let f(x) and g(x) be two function. One of the functions to be chosen and first function and other as second function by using the rule ILATE I- Inverse function Il- Logarithmic function Ill- Algebraic Function IV- Trigonometrical Function V- Exponential Function If we take fas the first function and g as the second function, then this formula may be stated as follows: “The integral of the product of two functions = (first function) x (integral of the second function) - Integral of [(differential coefficient of the first function) x (integral of the second function) ]” J fig (dx = f fs (x) dv — fe fi fec dx) dx DEFINITE INTEGRALS FUNDAMENTAL THEOREM OF CALCULUS r) AREA FUNCTION- The J f(dax is defined as the area bounded by the curve y = f(x), the x-axis and the a ordinates x=a and x=b. First fundamental theorem of integral calculus Let the area function be defined by A(x) = J “f) dx for all x >a, where a : the function fis assumed to be continuous dn [a, b]. Then A’ (x) =f (x) for all xe fa, b). Second fundamental theorem of integral calculus Let fbe a continuous function of .. defined on the closed interval [a, 6] ancl a let F be another function such that 7 dx F(x) = f(x) for all x in the domain of f-then fireo dx =[F(x) +C]? = F(b)- F(a). This is called the definite integral of f over the range [a, b], where a and b are called the limits of integration, @ being the lower limit and & the upper limit Scanned with CamScanner