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1999

JEE Main 1999 Mathematics question on Definite Integrals and Areas

∫π/43π/4dx1+cos⁡x\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x} is equal to
  1. A.
    2
  2. B.
    -2
  3. C.
    12\frac{1}{2}
  4. D.
    −12-\frac{1}{2}
Show answer and solution

Correct option: A

Solution

Use 1 + cos x = 2cos²(x/2), so the integral becomes ∫ dx / (2cos²(x/2)) = (1/2)∫ sec²(x/2) dx = tan(x/2) evaluated from π/4 to 3π/4.

tan(3π/8) − tan(π/8). Using the identity, tan(3π/8) = cot(π/8) = 1/tan(π/8). With tan(π/8) = √2 − 1, we get tan(3π/8) = √2 + 1, so the result is (√2+1) − (√2−1) = 2.

[1999]

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