int frac{dx}{sin x + cos x} = | vedclass

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(C) To evaluate the integral $I = \int \frac{dx}{\sin x + \cos x}$,we multiply and divide by $\sqrt{2}$:$I = \frac{1}{\sqrt{2}} \int \frac{dx}{\frac{1

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}} \log 	an \left( \frac{\pi}{8} + \frac{x}{2} ight) + c$

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(C) To evaluate the integral $I = \int \frac{dx}{\sin x + \cos x}$,we multiply and divide by $\sqrt{2}$:$I = \frac{1}{\sqrt{2}} \int \frac{dx}{\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x}$Using the identity $\sin(x + \frac{\pi}{4}) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x$,we get:$I = \frac{1}{\sqrt{2}} \int \frac{dx}{\sin(x + \frac{\pi}{4})}$$I = \frac{1}{\sqrt{2}} \int \csc(x + \frac{\pi}{4}) dx$Using the standard integral formula $\int \csc 	heta d	heta = \log |	an(\frac{	heta}{2})| + c$,we have:$I = \frac{1}{\sqrt{2}} \log \left| 	an \left( \frac{x + \frac{\pi}{4}}{2} ight) ight| + c$$I = \frac{1}{\sqrt{2}} \log 	an \left( \frac{x}{2} + \frac{\pi}{8} ight) + c$

$\int \frac{dx}{\sin x + \cos x} = $

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