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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