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Evaluate the definite integral ∫_(π/4)^(3π/4) (dx)/(1+cos x). What is its value?

1. -1
2. -2
3. 2
4. 4

Correct answer: 2

Solution

1+cos x = 2cos^2(x/2), so the integrand is (1/2)sec^2(x/2) and the antiderivative is tan(x/2). Evaluating from pi/4 to 3pi/4 gives tan(3pi/8)-tan(pi/8) = (1+sqrt(2))-(sqrt(2)-1) = 2.

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