Notes On Integrals involving Exponential Functions - CBSE Class 12 Maths
Integrals of some functions whose integrals are unknown. i) âˆ« log x dx ii) âˆ« sin-1 dx iii) âˆ« cos-1 dx iv) âˆ« tan-1 x dx sol 1) : âˆ« log x dx â‡’ âˆ« log x . 1 dx âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx âˆ« log x . 1 dx = log x âˆ« 1 dx - âˆ« [d/dx log x âˆ« 1 dx]dx = log x . x - âˆ« [1/x . x] dx = log x .x - âˆ« 1 dx = log x . x - x + C âˆ« log x . 1 dx = x(log x - 1) + C ii) âˆ« sin-1 dx Sol : âˆ« sin-1 . 1 dx âˆ« sin-1 . 1 dx = sin-1 x âˆ« 1 dx - âˆ« [d/dx sin-1 x âˆ« 1 dx ] dx                      = sin-1 x . x - âˆ«[1/âˆš(1-x2) . x]dx âˆ« sin-1 . 1 dx = = x sin-1 x + âˆš(1-x2) +  C iii) âˆ« cos-1 x dx Sol : âˆ« cos-1 x .1 dx âˆ« cos-1 x .1 dx = cos-1 x âˆ« 1dx - âˆ«[d/dx cos-1 x âˆ« 1 dx]dx Formula: âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx = cos-1 x . x - âˆ« [-1/âˆš(1-x2) . x] dx = cos-1 x . x + âˆ« [1/âˆš(1-x2) . x] dx âˆ´ âˆ« [1/âˆš(1-x2) . x] dx = âˆ« 1/âˆšt (dt/-2)   = -1/2. 2âˆšt = -âˆšt = -âˆš(1-x2) âˆ« cos-1 x dx = x cos-1 x - âˆš(1-x2) + C iv) tan-1 x dx Sol : âˆ« tan-1x .1 dx âˆ« tan-1x .1 dx = tan-1 x âˆ« 1 dx - âˆ«[d/dx tan-1x âˆ« 1 dx ]dx Formula: âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx = tan-1 x . x - âˆ«[1/(1+x2) . x] dx âˆ´ âˆ« [1/(1+x2) . x] dx = âˆ« 1/t (dt/2)  = 1/2 log|t| = 1/2 log|1+x2| âˆ« tan-1 x dx = x tan-1 x  - 1/2 log|1+x2| + C The integrand is the product of an exponential function. Consider âˆ« ex[f(x) + f '(x)] dx         = âˆ« [exf(x) + ex f '(x)] dx         = âˆ« ex f(x) dx + âˆ« ex  f '(x) dx         = f(x) âˆ« ex dx - âˆ«[d/dx[f(x)] âˆ« ex dx] dx + âˆ« ex  f '(x) dx          = f(x) . ex - âˆ« f '(x) ex  dx + âˆ« ex  f '(x) dx             âˆ« ex[f(x) + f '(x)] dx = exf(x) + C

#### Summary

Integrals of some functions whose integrals are unknown. i) âˆ« log x dx ii) âˆ« sin-1 dx iii) âˆ« cos-1 dx iv) âˆ« tan-1 x dx sol 1) : âˆ« log x dx â‡’ âˆ« log x . 1 dx âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx âˆ« log x . 1 dx = log x âˆ« 1 dx - âˆ« [d/dx log x âˆ« 1 dx]dx = log x . x - âˆ« [1/x . x] dx = log x .x - âˆ« 1 dx = log x . x - x + C âˆ« log x . 1 dx = x(log x - 1) + C ii) âˆ« sin-1 dx Sol : âˆ« sin-1 . 1 dx âˆ« sin-1 . 1 dx = sin-1 x âˆ« 1 dx - âˆ« [d/dx sin-1 x âˆ« 1 dx ] dx                      = sin-1 x . x - âˆ«[1/âˆš(1-x2) . x]dx âˆ« sin-1 . 1 dx = = x sin-1 x + âˆš(1-x2) +  C iii) âˆ« cos-1 x dx Sol : âˆ« cos-1 x .1 dx âˆ« cos-1 x .1 dx = cos-1 x âˆ« 1dx - âˆ«[d/dx cos-1 x âˆ« 1 dx]dx Formula: âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx = cos-1 x . x - âˆ« [-1/âˆš(1-x2) . x] dx = cos-1 x . x + âˆ« [1/âˆš(1-x2) . x] dx âˆ´ âˆ« [1/âˆš(1-x2) . x] dx = âˆ« 1/âˆšt (dt/-2)   = -1/2. 2âˆšt = -âˆšt = -âˆš(1-x2) âˆ« cos-1 x dx = x cos-1 x - âˆš(1-x2) + C iv) tan-1 x dx Sol : âˆ« tan-1x .1 dx âˆ« tan-1x .1 dx = tan-1 x âˆ« 1 dx - âˆ«[d/dx tan-1x âˆ« 1 dx ]dx Formula: âˆ« f(x) g(x) dx = f(x) âˆ« g(x) dx - âˆ« [ f '(x) âˆ« g(x) dx]  dx = tan-1 x . x - âˆ«[1/(1+x2) . x] dx âˆ´ âˆ« [1/(1+x2) . x] dx = âˆ« 1/t (dt/2)  = 1/2 log|t| = 1/2 log|1+x2| âˆ« tan-1 x dx = x tan-1 x  - 1/2 log|1+x2| + C The integrand is the product of an exponential function. Consider âˆ« ex[f(x) + f '(x)] dx         = âˆ« [exf(x) + ex f '(x)] dx         = âˆ« ex f(x) dx + âˆ« ex  f '(x) dx         = f(x) âˆ« ex dx - âˆ«[d/dx[f(x)] âˆ« ex dx] dx + âˆ« ex  f '(x) dx          = f(x) . ex - âˆ« f '(x) ex  dx + âˆ« ex  f '(x) dx             âˆ« ex[f(x) + f '(x)] dx = exf(x) + C

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