Question

49. Evaluate ſsin (log x) dx.​

Answer 1

Answer :

Rule -

∫ u v dx

= u ∫ v dx - ∫ { d/dx (u) ∫ v dx } dx

Now,

∫ sin (logx) dx

= sin (logx) × ∫ dx - ∫ { d/dx sin (logx) ∫dx } dx

= sin (logx) × x - ∫ { cos (logx) × 1/x × x } dx + c,

c = integral constant

= x sin (logx) - ∫ cos (logx) dx + c

= x sin (logx) - [ cos (logx) ∫ dx - ∫ { d/dx cos (logx) ∫ dx } ] + c

= x sin (logx) - x cos (logx)

+ ∫ { - sin (logx) × 1/x × x } dx + c

= x {sin (logx) - cos (logx)}

- ∫ sin (logx) dx + c

⇒ 2 ∫ sin (logx) dx

= x {sin (logx) - cos (logx)} + c

⇒ ∫ sin (logx) dx

= x/2 {sin (logx) - cos (logx) } + c/2

.

hope it helps