$Evaluate \int_0^{\pi/2} \frac{1 - \cot x}{\csc x + \cos x} \, dx:$

Question

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The given integral is: $$I = \int_{0}^{\frac{\pi}{2}} \frac{1 - \cot x}{\csc x + \cos x} dx.$$ Analyze the symmetry of the integral. The limits of the integral are symmetric about $\frac{\pi}{4}$, and the integrand contains terms that involve trigonometric functions $\sin x$, $\cos x$, $\cot x$, and $\csc x$. Specifically, consider the property: $$f(x) = -f\left(\frac{\pi}{2} - x\right).$$ For the integrand: $$f(x) = \frac{1 - \cot x}{\csc x + \cos x}.$$ Using the trigonometric substitutions: $$\cot\left(\frac{\pi}{2} - x\right) = \tan x, \quad \csc\left(\frac{\pi}{2} - x\right) = \sec x, \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x,$$ we find that the integrand satisfies the property: $$f(x) + f\left(\frac{\pi}{2} - x\right) = 0.$$ Since $f(x)$ is odd with respect to $x = \frac{\pi}{4}$, the integral over the symmetric interval $[0, \frac{\pi}{2}]$ evaluates to 0. Thus: $$ I = 0. $$

Answer

Answer

$ \frac{\pi}{4} $

Answer

<div>$ \infty $</div>

Answer

$ \frac{\pi}{12} $

List I		List II
A.	\(\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{7}{2}}x}{\sin^{\frac{7}{2}}+\cos^{\frac{7}{2}}}dx\)	I.	\(\frac{\pi}{4}-\frac{1}{2}\)
B.	\(\int\limits_0^{\pi}\frac{x\sin x}{1+\cos^2x}dx\)	II.	0
C.	\(\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}x\cos x\ dx\)	III.	\(\frac{\pi}{4}\)
D.	\(\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^2x\ dx\)	IV.	\(\frac{\pi^2}{4}\)

\(Evaluate \int_0^{\pi/2} \frac{1 - \cot x}{\csc x + \cos x} \, dx:\)

The Correct Option is A

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