INTEGRATION BY PARTS EXAMPLES AND SOLUTIONS

Integrate the following :

(1)  x e^-x

(2)  x cos x

(3)  x cosec²x

(4)  x sec x tan x dx

(5)  2 x e^3x

(6)   x² e ^2x

(7)  x⁵ e^x²

Problem 1 :

x e^-x

Solution :

∫x e^-x dx

∫ u dv  =  u v - ∫ v du

u  =  x            dv  =  e^-x

du  =  dx         v  =  -e^-x

=  ∫x e^-x dx

=  x ( -e^-x) - ∫ (- e^-x) dx

= - x e^-x + ∫ e^-x dx    

= - x e^-x - e^-x + C

= -e^-x (x + 1) + C

Problem 2 :

x cos x

Solution :

∫ x cos x dx

∫ u dv  =  u v - ∫v du

u  =  x           dv  =  cos x

du  =  dx          v  =  sin x

=  ∫x cos x dx

=  x (sin x)  - ∫ sin x  dx

=  x (sin x)  - (- cos x) + C

=  x  sin x + cos x + C

Problem 3 :

x cosec²x

Solution :

∫x cosec²x dx

∫ u dv = u v - ∫ v du

u  =  x           dv  =  cosec ²x

du  =  dx         v  =  -cot x

=  ∫x cosec ²x dx

=   x(-cot x) - ∫(-cot x) dx

=  -x cot x + ∫ (cos x/sin x) dx

=  - x cot x + log (sin x) + C

Problem 4 :

x sec x tan x dx

Solution :

∫x sec x tan x  dx

∫ u dv = u v - ∫ v du

u = x           dv = sec x tan x

du = dx         v = sec x

= ∫ x sec x tan x  dx

=  x (sec x) - ∫sec x dx

=  x sec x - log (sec x + tan x) + C

Problem 5 :

2 x e^3x

Solution :

∫ 2 x e^3x dx

u  =  x             dv  =  e^3x

du  =  dx          v = e^3x/3

=  2 [x (e^3x/3)- (e^3x/3) dx}

=  2[(x/3) [e^3x] - (1/3)∫[e^3x] dx

=  (2x/3) (e^3x) - (2/3)[e^(3x)/3] + C

=  (2x/3) (e^3x) - (2/9)(e^(3x)) + C

=  (2/3) e^3x [x - (1/3)] + C

Problem 6 :

 x² e ^2x

Solution :

∫ x² e^2x dx

u  =  x²                  dv  =  e^2x

du  =  2x dx           v  =  e^2x/2

=  { x²e^2x/2 - ∫ [e^2x/2] 2 x dx}

=  (x²/2)e^2x - ∫ [x e^2x] dx

u  =  x            dv  =  e^2x

du  =  dx          v  =  e^2x/2

=  x [e^2x/2]  - ∫ [e^2x/2] dx

=  (x/2) [e^2x] - (1/2)∫ [e^2x] dx

=  (x/2) [e^2x] - (1/2)[e^2x/2] + C

=  (x/2) [e^2x] - (1/4)[e^2x] + C

=  (1/2) (e^2x)[x - (1/2)] + C

Problem 7 :

x⁵ e^x²

Solution :

∫ x⁵ e^x²dx = ∫(x²)² x e^x² dx

let t  =  x²

dt  =  2 x dx

x dx  =  dt//2

∫(x²)² x e^x² dx   =  ∫t² e^t (dt/2)

=  (1/2)∫t² e^t dt

u  =  v                     dv  =  e^t

du  =  2t dt           v  =  e^t

∫ u dv = uv - ∫ vdu

=  (1/2) [t²e^t - ∫e^t  (2 t) dt]

=  (1/2) [t² e^t - 2 ∫ t e^t dt]

u  =  t             dv  =  e^t

du  =  dt           v  =  e^t

=  (1/2) {t² e^t - 2 [t e^t- ∫ e^t dt]}

=  (1/2) t² e^t - [t e^t- e^t] + C

=  [(1/2) t² e^t] - t e^t + e^t + C

=  (1/2)e^t [t²- 2 t + 2]+ C

=  (1/2)e^x² [(x²)²- 2 x² + 2] + C

=  (1/2)e^x² [x⁴- 2 x²+ 2]+ C

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