First slide

Evaluation of definite integrals

Question

Let $I = \int_{0}^{π} \frac{\cos x}{\sqrt{1 - \sin^{2} x}} d x$
Statement-1: $I = π$
Statement-2: The integrand can be expressed as $X_{[0, π / 2]} X_{[π / 2, π]}; X_{A}$ being the characteristic function of A.

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

$\frac{\cos x}{\sqrt{1 - \sin^{2} x}} = \frac{\cos}{| \cos x |} = \{\begin{matrix} 1 & x \in [0, π / 2] \\ - 1 & , x \in (π / 2, π) \end{matrix}$
$\int_{0}^{π} \frac{\cos x}{\sqrt{1 - \sin^{2} x}} = \int_{0}^{π / 2} d x - \int_{π / 2}^{π} d x = 0$

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