Find ∫ sin⁡x log⁡(cos⁡x) dx.

Category: QuestionsFind ∫ sin⁡x log⁡(cos⁡x) dx.
Editor">Editor Staff asked 11 months ago

Find ∫ sin⁡x log⁡(cos⁡x) dx.
 
(a) cos⁡x (log⁡(sin⁡x)-1)+C
 
(b) sin⁡x (log⁡(cos⁡x)+1)+C
 
(c) cos⁡x (log⁡(cos⁡x)-1)+C
 
(d) cos⁡x (log⁡(cos⁡x)-1)+C
 
The question was asked in final exam.
 
My question is taken from Integration by Parts in section Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Right choice is (c) cos⁡x (log⁡(cos⁡x)-1)+C
 
To explain I would say: Let cos⁡x=t
 
Differentiating w.r.t x, we get
 
-sin⁡x dx=dt
 
sin⁡x dx=-dt
 
∴∫ sin⁡x log⁡(cos⁡x) dx=∫ -log⁡t dt
 
Using ∫ u.v dx=u∫ v dx-∫ u’ (∫ v dx) , we get
 
∫ -log⁡t dt=-log⁡t ∫ 1 dt-∫ (-log⁡t)’ ∫ 1 dt
 
=-t log⁡t+∫ dt
 
=-t log⁡t+t
 
=t(log⁡t-1)
 
Replacing t with cos⁡x, we get
 
∫ sin⁡x log⁡(cos⁡x) dx=cos⁡x (log⁡(cos⁡x)-1)+C