Evaluate: ∫ e^x [ (1 + sin x) / (1 + cos x) ] dx.
Answer and explanation
Correct answer: C. e^x tan(x/2) + C
- Step 1: Rewrite (1 + sin x)/(1 + cos x) = [1 + 2 sin(x/2)cos(x/2)] / [2 cos^2(x/2)].
- Step 2: = (1/2) sec^2(x/2) + tan(x/2).
- Step 3: Standard form ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C, where f(x) = tan(x/2) and f'(x) = (1/2) sec^2(x/2).
- Step 4: Integral = e^x tan(x/2) + C.
Why not the others: The other options (e^x sec(x/2) + C, e^x cos(x/2) + C, (1/2) e^x tan(x/2) + C) result from applying the power rule incorrectly, forgetting to differentiate lower-order terms, or arithmetic substitution errors.
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