Published by:
CGP EDU Academic Team
Published on: August 13, 2026
If φ (x) = – 
log cos y dy, prove that
φ (x) = 2 φ 
–2 φ 
–x log 2
Hence or otherwise show that φ 
= 
log 2
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. We have,
φ (x) = – 
∴ φ 
= – 
and, φ 
= – 
Let I = φ 
– φ 
Then,
I = – 
+ 
⇒ I = 
+ 
⇒ I = 
... (i)
⇒ I = 
dy
⇒ I = 
dy
⇒ I = 
...(ii)
Adding (i) and (ii), we get
2I = 
= 
= 
– 
= 
= 
= x log 2, where
y = 
+ 
= 
+ x log 2
= – 
+ x log 2
= – 
+ x log 2
= φ (x) + x log 2
= 2 
= φ (x) + x log 2
⇒ φ (x) = 2 φ 
–2 φ 
– x log 2
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