Integrate the following :
(1) x e-x
(2) x cos x
(3) x cosec2x
(4) x sec x tan x dx
(5) 2 x e3x
(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 e3x
Solution :
∫ 2 x e3x dx
u = x dv = e3x
du = dx v = e3x/3
= 2 [x (e3x/3)- (e3x/3) dx}
= 2[(x/3) [e3x] - (1/3)∫[e3x] dx
= (2x/3) (e3x) - (2/3)[e^(3x)/3] + C
= (2x/3) (e3x) - (2/9)(e^(3x)) + C
= (2/3) e3x [x - (1/3)] + C
Problem 6 :
x2 e 2x
Solution :
∫ x2 e2x dx
u = x2 dv = e2x
du = 2x dx v = e2x/2
= { x²e2x/2 - ∫ [e2x/2] 2 x dx}
= (x²/2)e2x - ∫ [x e2x] dx
u = x dv = e2x
du = dx v = e2x/2
= x [e2x/2] - ∫ [e2x/2] dx
= (x/2) [e2x] - (1/2)∫ [e2x] dx
= (x/2) [e2x] - (1/2)[e2x/2] + C
= (x/2) [e2x] - (1/4)[e2x] + C
= (1/2) (e2x)[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 = et
du = 2t dt v = et
∫ u dv = uv - ∫ vdu
= (1/2) [t²et - ∫et (2 t) dt]
= (1/2) [t² et - 2 ∫ t et dt]
u = t dv = et
du = dt v = et
= (1/2) {t² et - 2 [t et- ∫ et dt]}
= (1/2) t² et - [t et- et] + C
= [(1/2) t² et] - t et + et + C
= (1/2)et [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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