int_π/4^3π/4 dx/(1 + cos x) is equal to — JEE Main 2017 Mathematics PYQ

JEE Main 2017

\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}

is equal to

A.
$-2$
B.
$2$
C.
$4$
D.
$-1$

Show answer and solution

Correct option: B

Solution

1 + cos x = 2cos²(x/2), so the integral becomes ∫(π/4 to 3π/4) 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). Let t = tan(π/8); the result is 1/t - t = (1 - t²)/t = 2/tan(π/4) · ... more directly: tan(π/4) = 2tan(π/8)/(1-tan²(π/8)) = 1, so (1 - t²)/t = 2, giving the answer 2.

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