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Class 12 Integrals · Short Answer2 Marksabsolute_value_branches

Class 12 Maths · Integrals · Short Answer #54

Evaluate \(\int e^x\sqrt{1+\sin2x}/(1+\cos2x)\,dx\) on intervals where cos x≠0 and sin x+cos x has a fixed nonzero sign. State both sign cases.

✅ Step-by-Step Solution

Final Answer: s e^x sec x/2+C, where s=sign(sin x+cos x).
  1. √(1+sin2x)=|sin x+cos x| and 1+cos2x=2cos²x. Set s=sign(sin x+cos x), constant on the chosen interval.
  2. The integrand becomes (s/2)e^x(sec x+sec x tan x)=(s/2)d(e^x sec x)/dx.
  3. The integral is e^x sec x/2+C when sin x+cos x>0 and −e^x sec x/2+C when it is negative. To extend across a zero, choose adjacent constants to make the primitive continuous.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] √(1+sin2x)=|sin x+cos x| and 1+cos2x=2cos²x. Set s=sign(sin x+cos x), constant on the chosen interval.
  • [1 mark] The integrand becomes (s/2)e^x(sec x+sec x tan x)=(s/2)d(e^x sec x)/dx.
  • [0.5 marks] The integral is e^x sec x/2+C when sin x+cos x>0 and −e^x sec x/2+C when it is negative. To extend across a zero, choose adjacent constants to make the primitive continuous.

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