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JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Definite Integrals

If

\int_{0}^{\frac{π}{2}} \frac{c o t x}{c o t x + c o s ec x} d x = m (π + n)

, then

m \cdot n

is equal to

(A)1
(B) $\frac{1}{2}$
(C) $- \frac{1}{2}$
(D)-1✓ Correct

Answer: (D)

Solution

Simplify the integrand first:

\frac{cot x}{cot x + cosec x} = \frac{cos x}{cos x + 1}

. Using

cos x = 2 cos^{2} \frac{x}{2} - 1

and

1 + cos x = 2 cos^{2} \frac{x}{2}

, this becomes

1 - \frac{1}{2} sec^{2} \frac{x}{2}

. Integrating from

0

to

\frac{π}{2}

,

\int_{0}^{π /2} (1 - \frac{1}{2} sec^{2} \frac{x}{2}) d x = [x - tan \frac{x}{2}]_{0}^{π /2} = \frac{π}{2} - 1

. Matching

\frac{π}{2} - 1 = m (π + n)

gives

m = \frac{1}{2}

and

n = - 2

, so

m \cdot n = - 1

.

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option

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