Presentation on theme: "Indefinite integrals Definition: if f(x) be any differentiable function of such that d/dx f( x ) = f(x)Is called an anti-derivative or an indefinite integral."— Presentation transcript:
1 Indefinite integralsDefinition: if f(x) be any differentiable function of such that d/dx f( x ) = f(x)Is called an anti-derivative or an indefiniteintegral of f(x) we write f(x) = ∫ f (x) dx. The symbol∫ dx isused as a symbol of operation which means integratingw.r.t. x.
2 1.∫ k f(x) dx = k ∫ f(x) dx, where k is a constant In all the above formulas, c is the constant of integration.Some standard results on integration1.∫ k f(x) dx = k ∫ f(x) dx, where k is a constant2. ∫ ( f(x) ± g(x))dx = ∫ f(x)dx ± ∫ g(x)dx
5 METHODS OF INTEGRATION It was based on inspection, i.e,. On the searchof a function F whose derivative is f which ledus to the integral of f . However, this method, whichdepends on inspection , is not very suitable formany functions.Hence, we need to develop additional techniquesor methods for finding the integrals by reducingThem into standard forms.
6 Following are the method of integration:Integration by substitutionIntegration by partIntegration by partial fractions
7 Integration by substitution This method is used when basic rules ofIntegration given above, are not directlyapplicable. In such cases, the integrationis transformed (by suitable substitutionof given variable by a new variable) into aform to which basic rules of integration areapplicable
8 ∫√t dt = ∫t½ dt = 2/3t³⁄² + C Equation of integrals √15 + logx dx x put15 + logx = t1/x dx = dtSubstituting in given question, we get∫√t dt = ∫t½ dt = 2/3t³⁄² + C
9 Conti……= 2/3 (15 + logx)³⁄² + CWhere c = Constant of integration.
10 ∫ Integration by parts ∫ ∫ ∫ This method is based on product rule differentiation.According to this method, the integralof product of Two functionsf(x). g(x) is given by∫g(x) dx - f ’(x) ( g(x)dx) dx∫∫∫F(x) . g(x) dx = f(x)
11 ∫ I . II dx = I ∫IIdx - ∫(d(I)/dx) dx Taking f(x) as the first function (I) andg(x) as the second (II) the above equationcan be Stated as∫ I . II dx = I ∫IIdx - ∫(d(I)/dx) dx
12 Rule for the proper chose of the first function The first function to be selected may be the one which comes first in the order ILATE whereI means inverse trigonometric functionL means logarithmic functionA means algebraicT means trigonometricE means exponential.
13 ∫x² e dxxxTake x² as the function and e as the second function and integrating by parts, we getI = x² e - ∫2x exXdx
14 I = x² e – 2 [xe - ∫e dx] = x² e – 2xe + 2e + C, Again apply by parts in second term, we getxxI = x² e – 2 [xe - ∫e dx]XX= x² e – 2xe + 2e + C,Where c. is the constant of integrationX
15 Integration by partial fraction If f(x) and g(x) are two polynomials,then f(x)/g(x) is called a rational algebraicFunction of x.Any rational f (x)/ g(x) can be expressed asthe sum of rational functions each having asimple factor of g(x). Each such fraction iscalled a partial fraction.
16 3x1. ∫dx(x-1) (x+2)AB=+X-1X + 23x = A (X + 2) + B (X + 1)
17 Put,,X = we get3 = 3AA = 1Put,x = -2 we get- 6 = - 3BB = 2
18 ∫ ∫ ∫ = (x+2) (x-1) 2 + 1 X - 1 3x X-1 X + 2 3x dx + 2 X + 2 Now integrating, we get3x21X - 1∫∫dx =∫dx + 2( x - 1 ) ( x + 2)X + 2