The integral of sqrt{1 + 2 cot x (cot x + csc | vedclass

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Question

(B) Let $I = \int \sqrt{1 + 2 \cot x (\cot x + \csc x)} \, dx$.Using the identity $1 + \cot^2 x = \csc^2 x$,we can rewrite the expression inside the s

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) Let $I = \int \sqrt{1 + 2 \cot x (\cot x + \csc x)} \, dx$.Using the identity $1 + \cot^2 x = \csc^2 x$,we can rewrite the expression inside the square root:$1 + 2 \cot^2 x + 2 \cot x \csc x = \csc^2 x + \cot^2 x + 2 \cot x \csc x$.This is a perfect square: $(\csc x + \cot x)^2$.So,$I = \int \sqrt{(\csc x + \cot x)^2} \, dx = \int (\csc x + \cot x) \, dx$.The integral of $\csc x$ is $\ln|\csc x - \cot x|$ and the integral of $\cot x$ is $\ln|\sin x|$.Alternatively,$\csc x + \cot x = \frac{1 + \cos x}{\sin x} = \frac{2 \cos^2(x/2)}{2 \sin(x/2) \cos(x/2)} = \cot(x/2)$.The integral of $\cot(x/2)$ is $2 \ln|\sin(x/2)| + c$.

The integral of $\sqrt{1 + 2 \cot x (\cot x + \csc x)}$ with respect to $x$ is:

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