INTEGRATION OF e POWER ax Sin bx or Cos bx

Question 1 :

Integrate the following with respect to x

 eax cosbx

Solution :

 ∫ eax cosbx dx  =  eax/(a2 + b2) (a cos bx + b sin bx)

Question 2 :

Integrate the following with respect to x

 e2x sin x

Solution :

 ∫ eax sin bx dx  =  eax/(a2 + b2) (a sin bx - b cos bx)

 ∫  e2x sin x dx 

a  =  2 and b = 1

  =  e2x/(22 + 12) (2 sin x - 1 cos x) + c

  =  (e2x/5) (2 sin x - 1 cos x) + c

Question 3 :

Integrate the following with respect to x

 e-x cos 2x

Solution :

 ∫ eax cosbx dx  =  eax/(a2 + b2) (a cos bx + b sin bx)

 ∫  e-x cos 2x dx 

a  =  -1 and b = 2

  =  e-x/((-1)2 + 22) (-1 cos 2x + 2 sin 2x) + c

  =  (e-x/5) (-cos 2x + 2 sin 2x) + c

  =  (e-x/5) (2 sin 2x - cos 2x) + c

Question 4 :

Integrate the following with respect to x

 e-3x sin 2x

Solution :

 ∫ eax sin bx dx  =  eax/(a2 + b2) (a sin bx - b cos bx)

 ∫ e-3x sin 2x dx 

a  =  -3 and b = 2

  =  e-3x/((-3)2 + 12) (-3 sin x - 2 cos x) + c

  =  (e-3x/10) (-3 sin x - 2 cos x) + c

  =  (-e-3x/10) (3 sin x + 2 cos x) + c

Question 5 :

Integrate the following with respect to x

 e-4x sin 2x

Solution :

 ∫ eax sin bx dx  =  eax/(a2 + b2) (a sin bx - b cos bx)

 ∫ e-4x sin 2x dx 

a  =  -4 and b = 2

  =  e-4x/((-4)2 + 12) (-4 sin x - 2 cos x) + c

  =  (e-4x/17) (-4 sin x - 2 cos x) + c

  =  (-e-4x/17) (4 sin x + 2 cos x) + c

Question 6 :

Integrate the following with respect to x

 e-3x cos x

Solution :

 ∫ eax cosbx dx  =  eax/(a2 + b2) (a cos bx + b sin bx)

 ∫  e-3x cos x dx 

a  =  -3 and b = 1

  =  e-3x/((-3)2 + 12) (-3 cos x + 1 sin x) + c

  =  (e-3x/10) (-3 cos x + 1 sin x) + c

  =  (e-3x/10) (1 sin x - 3 cos x) + c

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