int sqrt{1 + cos x} , dx equals | vedclass

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Question

(A) We know that $1 + \cos x = 2 \cos^2 \frac{x}{2}$.Substituting this into the integral,we get:$I = \int \sqrt{2 \cos^2 \frac{x}{2}} \, dx$$I = \sqrt

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$2\sqrt{2} \sin \frac{x}{2} + c$

Vedclass · Answer

(A) We know that $1 + \cos x = 2 \cos^2 \frac{x}{2}$.Substituting this into the integral,we get:$I = \int \sqrt{2 \cos^2 \frac{x}{2}} \, dx$$I = \sqrt{2} \int \cos \frac{x}{2} \, dx$Using the integration formula $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + c$,where $a = \frac{1}{2}$:$I = \sqrt{2} \cdot \frac{\sin(x/2)}{1/2} + c$$I = 2\sqrt{2} \sin \frac{x}{2} + c$.

$\int \sqrt{1 + \cos x} \, dx$ equals

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