Published by:
CGP EDU Academic Team
Published on: August 14, 2026
If I (u) = 
(1 –2u cos x + u 2 ) dx, then prove that
(i) I(u) I(–u) (ii) I(u) + I (–u) = I(u 2 )
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. (i) We have,
I(u) = 
(1 –2u cos x + u 2 ) dx
⇒ I (u) = 
(1 –2u cos ( π –x) + u 2 ) dx
⇒ I(u) = 
(1 + 2u cos x + u 2 ) dx
⇒ I(u) = I (–u)
(ii) We have,
I(u) + I (–u)
= 
[log (1–2u cos x + u 2 ) + log (1+ 2u cos x + u 2 )] du
= 
log {(1+ u 2 ) 2 –4u 2 cos 2 x} dx
= 
log {1 + u 4 –2u 2 (2 cos 2 x –1)} dx
= 
log (1+ u 4 –2u 2 cos 2x) dx
= 
log (1 + u 4 –2u 2 cos t) dt, where 2x = t
= 
× 2 
= 
log (1 + u 4 –2u 2 cos t) dt = I(u 2 )
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