JEEOlympiad
Evaluate I = \int_0^{\pi} \ln\!\big(5 - 4\cos x\big)\,dx and give its…
Question
Evaluate I = \int_0^{\pi} \ln\!\big(5 - 4\cos x\big)\,dx and give its value rounded to 4 decimal places. (Note 5-4\cos x = 1 - 2(2)\cos x + 2^2.)
✓ Verified answer: 4.3552checked by our engine — not a guess
Step-by-step solution
Recognize 5-4\cos x = |2 - e^{ix}|^2 = (1-2r\cos x + r^2) with r=2.
Use the Poisson-type result \int_0^{\pi}\ln(1-2r\cos x + r^2)\,dx = 0 for |r|\le1 and = 2\pi\ln|r| for |r|>1.
(Proof: write \ln(1-2r\cos x+r^2)=2\ln|r|+\ln(1-2r^{-1}\cos x+r^{-2}) for r>1; the second part integrates to 0 by the |r|<1 case.) With r=2: I = 2\pi\ln 2.
Numerically 2\pi\ln2 = 6.283185\times0.693147 = 4.355172.
Final answer4.3552
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