Notes On Properties of Indefinite Integral - CBSE Class 12 Maths
Property - I d/dx âˆ« f(x) dx = f(x) and âˆ« f '(x) dx = f(x) + C, where C is any arbitrary constant . Proof: Let G be any anti-derivative of f. d/dx[G(x)] = f(x) ....... (1) âˆ« f(x) dx = G(x) + C d/dx[âˆ« f(x) dx] = d/dx[G(x) + C] = d/dx[G(x)]  ['.' d/dx (C) = 0] = f(x)     [from (1)] âˆ´ d/dx[âˆ« f(x)dx] = f(x) We know that, âˆ« d/dx[f(x)]dx = âˆ« f ' (x) dx â‡’ âˆ« f '(x)dx = f(x) + C Here, C is called the constant of integration. Property - II: Two indefinite integrals having the same derivative lead to the same family of curves and they are equivalent. Proof: Let f and g be two functions such that, d/dx âˆ« f(x) dx = d/dx âˆ« g(x)dx â‡’d/dx âˆ« f(x)dx - d/dx âˆ« g(x)dx = 0 â‡’d/dx[âˆ«f(x)dx - âˆ«g(x)dx] = 0 d/dx(Constant) = 0 â‡’ âˆ« f(x)dx - âˆ« g(x) dx = C , arbitrary constant. âˆ« f(x)dx = âˆ« g(x) dx + C If C = C2 - C1, then âˆ« f(x)dx = âˆ« g(x) + C2 - C1 â‡’ âˆ« f(x)dx + C1 = âˆ« g(x) + C2 {âˆ« f(x)dx + C1,C2 âˆˆ R} and {âˆ« g(x)dx + C1,C2 âˆˆ R} are identical } âˆ« f(x)dx and âˆ« g(x)dx are equivalent. NOTE: The family of curves is {âˆ« f(x)dx + C1,C2 âˆˆ R} â‡”âˆ« f(x)dx {âˆ« g(x)dx + C1,C2 âˆˆ R} â‡” âˆ« g(x)dx Property - III If f and g are two functions, then âˆ« [f(x) + g(x)]dx = âˆ« f(x) dx + âˆ« g(x) dx Proof: d/dx âˆ« f(x) dx = f(x) d/dx âˆ« [f(x) + g(x)] dx = f(x) + g(x) .......(1) Consider d/dx [âˆ« f(x) dx + âˆ« g(x) dx] = d/dx âˆ« f(x) dx + d/dx âˆ« g(x) dx = f(x) + g(x).....(2)  d/dx âˆ« [f(x) + g(x)] dx =  d/dx [âˆ« f(x) dx + âˆ« g(x) dx]------- (3) âˆ« [f(x) + g(x)] dx = âˆ« f(x) dx+ âˆ« g(x) dx Hence, the integral of the sum of functions is equal to sum of the integral of the functions. Property - IV âˆ« k f(x) dx = k âˆ« f(x) dx, where k is any real number. Proof: d/dx âˆ« f(x) dx = f(x) d/dx âˆ« kf(x) dx = kf(x)-------(1) We know that , d/dx(k x) = k d/dx (x) d/dx[k âˆ« f(x)dx] = k d/dx âˆ« f(x) dx = k f(x)  ['.' d/dx âˆ« g(x) dx = g(x)] d/dx[k âˆ« f(x)dx] = k f(x).........(2) d/dx âˆ« k f(x)dx = d/dx[k âˆ« f(x)dx] d/dx âˆ« k f(x)dx - d/dx[k âˆ« f(x)dx] = 0 â‡’d/dx[âˆ« k f(x)dx - k âˆ« f(x)dx] = 0 Since the derivative of a constant is equal to zero , we have âˆ« kf(x)dx - k âˆ« f(x)dx = C âˆ« kf(x)dx = k âˆ« f(x)dx + C Hence, âˆ« k f(x)dx = k âˆ« f(x) dx. Property - V: âˆ«[k1f1(x) + k2f2(x) + k3f3(x) + .... +.... knfn(x)]dx = k1 âˆ« f1(x) dx + k2 âˆ« f2(x) dx + k3 âˆ« f3(x) dx + .... +.... kn âˆ« fn(x)]dx Proof: We combine properties III and IV for a finite number of functions such as f1,f2,f3,....,fn and for real numbers k1,k2,k3,....,kn. So, using properties III and IV, we have âˆ«[k1f1(x) + k2f2(x) + k3f3(x) + .... +.... knfn(x)]dx = k1 âˆ« f1(x) dx + k2 âˆ« f2(x) dx + k3 âˆ« f3(x) dx + .... +.... kn âˆ« fn(x)]dx .

Summary

Property - I d/dx âˆ« f(x) dx = f(x) and âˆ« f '(x) dx = f(x) + C, where C is any arbitrary constant . Proof: Let G be any anti-derivative of f. d/dx[G(x)] = f(x) ....... (1) âˆ« f(x) dx = G(x) + C d/dx[âˆ« f(x) dx] = d/dx[G(x) + C] = d/dx[G(x)]  ['.' d/dx (C) = 0] = f(x)     [from (1)] âˆ´ d/dx[âˆ« f(x)dx] = f(x) We know that, âˆ« d/dx[f(x)]dx = âˆ« f ' (x) dx â‡’ âˆ« f '(x)dx = f(x) + C Here, C is called the constant of integration. Property - II: Two indefinite integrals having the same derivative lead to the same family of curves and they are equivalent. Proof: Let f and g be two functions such that, d/dx âˆ« f(x) dx = d/dx âˆ« g(x)dx â‡’d/dx âˆ« f(x)dx - d/dx âˆ« g(x)dx = 0 â‡’d/dx[âˆ«f(x)dx - âˆ«g(x)dx] = 0 d/dx(Constant) = 0 â‡’ âˆ« f(x)dx - âˆ« g(x) dx = C , arbitrary constant. âˆ« f(x)dx = âˆ« g(x) dx + C If C = C2 - C1, then âˆ« f(x)dx = âˆ« g(x) + C2 - C1 â‡’ âˆ« f(x)dx + C1 = âˆ« g(x) + C2 {âˆ« f(x)dx + C1,C2 âˆˆ R} and {âˆ« g(x)dx + C1,C2 âˆˆ R} are identical } âˆ« f(x)dx and âˆ« g(x)dx are equivalent. NOTE: The family of curves is {âˆ« f(x)dx + C1,C2 âˆˆ R} â‡”âˆ« f(x)dx {âˆ« g(x)dx + C1,C2 âˆˆ R} â‡” âˆ« g(x)dx Property - III If f and g are two functions, then âˆ« [f(x) + g(x)]dx = âˆ« f(x) dx + âˆ« g(x) dx Proof: d/dx âˆ« f(x) dx = f(x) d/dx âˆ« [f(x) + g(x)] dx = f(x) + g(x) .......(1) Consider d/dx [âˆ« f(x) dx + âˆ« g(x) dx] = d/dx âˆ« f(x) dx + d/dx âˆ« g(x) dx = f(x) + g(x).....(2)  d/dx âˆ« [f(x) + g(x)] dx =  d/dx [âˆ« f(x) dx + âˆ« g(x) dx]------- (3) âˆ« [f(x) + g(x)] dx = âˆ« f(x) dx+ âˆ« g(x) dx Hence, the integral of the sum of functions is equal to sum of the integral of the functions. Property - IV âˆ« k f(x) dx = k âˆ« f(x) dx, where k is any real number. Proof: d/dx âˆ« f(x) dx = f(x) d/dx âˆ« kf(x) dx = kf(x)-------(1) We know that , d/dx(k x) = k d/dx (x) d/dx[k âˆ« f(x)dx] = k d/dx âˆ« f(x) dx = k f(x)  ['.' d/dx âˆ« g(x) dx = g(x)] d/dx[k âˆ« f(x)dx] = k f(x).........(2) d/dx âˆ« k f(x)dx = d/dx[k âˆ« f(x)dx] d/dx âˆ« k f(x)dx - d/dx[k âˆ« f(x)dx] = 0 â‡’d/dx[âˆ« k f(x)dx - k âˆ« f(x)dx] = 0 Since the derivative of a constant is equal to zero , we have âˆ« kf(x)dx - k âˆ« f(x)dx = C âˆ« kf(x)dx = k âˆ« f(x)dx + C Hence, âˆ« k f(x)dx = k âˆ« f(x) dx. Property - V: âˆ«[k1f1(x) + k2f2(x) + k3f3(x) + .... +.... knfn(x)]dx = k1 âˆ« f1(x) dx + k2 âˆ« f2(x) dx + k3 âˆ« f3(x) dx + .... +.... kn âˆ« fn(x)]dx Proof: We combine properties III and IV for a finite number of functions such as f1,f2,f3,....,fn and for real numbers k1,k2,k3,....,kn. So, using properties III and IV, we have âˆ«[k1f1(x) + k2f2(x) + k3f3(x) + .... +.... knfn(x)]dx = k1 âˆ« f1(x) dx + k2 âˆ« f2(x) dx + k3 âˆ« f3(x) dx + .... +.... kn âˆ« fn(x)]dx .

Previous
Next
âž¤