Integrals involving Exponential Functions

Integrals of some functions whose integrals are unknown.

i) ∫ log x dx

ii) ∫ sin-1 dx

iii) ∫ cos-1 dx

iv) ∫ tan-1 x dx


sol 1) : ∫ log x dx ⇒ ∫ log x . 1 dx

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

∫ log x . 1 dx = log x ∫ 1 dx - ∫ [d/dx log x ∫ 1 dx]dx

= log x . x - ∫ [1/x . x] dx

= log x .x - ∫ 1 dx

= log x . x - x + C

∫ log x . 1 dx = x(log x - 1) + C


ii) ∫ sin-1 dx

Sol : ∫ sin-1 . 1 dx

∫ sin-1 . 1 dx = sin-1 x ∫ 1 dx - ∫ [d/dx sin-1 x ∫ 1 dx ] dx

                     = sin-1 x . x - ∫[1/√(1-x2) . x]dx

∫ sin-1 . 1 dx = = x sin-1 x + √(1-x2) +  C



iii) ∫ cos-1 x dx

Sol : ∫ cos-1 x .1 dx

∫ cos-1 x .1 dx = cos-1 x ∫ 1dx - ∫[d/dx cos-1 x ∫ 1 dx]dx

Formula:

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

= cos-1 x . x - ∫ [-1/√(1-x2) . x] dx

= cos-1 x . x + ∫ [1/√(1-x2) . x] dx

∴ ∫ [1/√(1-x2) . x] dx = ∫ 1/√t (dt/-2)

  = -1/2. 2√t = -√t = -√(1-x2)

∫ cos-1 x dx = x cos-1 x - √(1-x2) + C



iv) tan-1 x dx

Sol : ∫ tan-1x .1 dx

∫ tan-1x .1 dx = tan-1 x ∫ 1 dx - ∫[d/dx tan-1x ∫ 1 dx ]dx

Formula:

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

= tan-1 x . x - ∫[1/(1+x2) . x] dx

∴ ∫ [1/(1+x2) . x] dx = ∫ 1/t (dt/2)

 = 1/2 log|t| = 1/2 log|1+x2|

∫ tan-1 x dx = x tan-1 x  - 1/2 log|1+x2| + C

The integrand is the product of an exponential function.

Consider ∫ ex[f(x) + f '(x)] dx

        = ∫ [exf(x) + ex f '(x)] dx

        = ∫ ex f(x) dx + ∫ ef '(x) dx

        = f(x) ∫ ex dx - ∫[d/dx[f(x)] ∫ ex dx] dx + ∫ ef '(x) dx

         = f(x) . ex - ∫ f '(x) e dx + ∫ ef '(x) dx

            ∫ ex[f(x) + f '(x)] dx = exf(x) + C

Summary

Integrals of some functions whose integrals are unknown.

i) ∫ log x dx

ii) ∫ sin-1 dx

iii) ∫ cos-1 dx

iv) ∫ tan-1 x dx


sol 1) : ∫ log x dx ⇒ ∫ log x . 1 dx

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

∫ log x . 1 dx = log x ∫ 1 dx - ∫ [d/dx log x ∫ 1 dx]dx

= log x . x - ∫ [1/x . x] dx

= log x .x - ∫ 1 dx

= log x . x - x + C

∫ log x . 1 dx = x(log x - 1) + C


ii) ∫ sin-1 dx

Sol : ∫ sin-1 . 1 dx

∫ sin-1 . 1 dx = sin-1 x ∫ 1 dx - ∫ [d/dx sin-1 x ∫ 1 dx ] dx

                     = sin-1 x . x - ∫[1/√(1-x2) . x]dx

∫ sin-1 . 1 dx = = x sin-1 x + √(1-x2) +  C



iii) ∫ cos-1 x dx

Sol : ∫ cos-1 x .1 dx

∫ cos-1 x .1 dx = cos-1 x ∫ 1dx - ∫[d/dx cos-1 x ∫ 1 dx]dx

Formula:

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

= cos-1 x . x - ∫ [-1/√(1-x2) . x] dx

= cos-1 x . x + ∫ [1/√(1-x2) . x] dx

∴ ∫ [1/√(1-x2) . x] dx = ∫ 1/√t (dt/-2)

  = -1/2. 2√t = -√t = -√(1-x2)

∫ cos-1 x dx = x cos-1 x - √(1-x2) + C



iv) tan-1 x dx

Sol : ∫ tan-1x .1 dx

∫ tan-1x .1 dx = tan-1 x ∫ 1 dx - ∫[d/dx tan-1x ∫ 1 dx ]dx

Formula:

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

= tan-1 x . x - ∫[1/(1+x2) . x] dx

∴ ∫ [1/(1+x2) . x] dx = ∫ 1/t (dt/2)

 = 1/2 log|t| = 1/2 log|1+x2|

∫ tan-1 x dx = x tan-1 x  - 1/2 log|1+x2| + C

The integrand is the product of an exponential function.

Consider ∫ ex[f(x) + f '(x)] dx

        = ∫ [exf(x) + ex f '(x)] dx

        = ∫ ex f(x) dx + ∫ ef '(x) dx

        = f(x) ∫ ex dx - ∫[d/dx[f(x)] ∫ ex dx] dx + ∫ ef '(x) dx

         = f(x) . ex - ∫ f '(x) e dx + ∫ ef '(x) dx

            ∫ ex[f(x) + f '(x)] dx = exf(x) + C

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