Question

the value of ∫cot^2xdx is________________

• A. log\(\sqrt{sin\,2x}\)\(+c\)
• B. log√\(\sqrt{sec\,2x}+c\)
• C. log\(\sqrt{cosec\,2x}+c\)
• D. log\(\sqrt{cos\,2x}+c\)

Answer 1

The Correct Option is A

To determine the value of the integral \(\int \cot^{2}x \, dx\), we can use trigonometric identities and integration techniques.

Start by expressing \(\cot^{2}x\) in terms of \(\csc^{2}x\):

\(\cot^{2}x = \csc^{2}x - 1\)

The integral becomes:

\(\int \cot^{2}x \, dx = \int (\csc^{2}x - 1) \, dx\)

This integral can be split into two separate integrals:

\(\int \cot^{2}x \, dx = \int \csc^{2}x \, dx - \int 1 \, dx\)

The integral of \(\csc^{2}x\) is \(-\cot x\), and the integral of \(1\) is \(x\). Therefore, the expression becomes:

\(-\cot x - x + C\) where \(C\) is the constant of integration.

Using the trigonometric identity \(\cot x = \frac{\cos x}{\sin x}\), rewrite the result:

\(-\frac{\cos x}{\sin x} - x + C\)

To further simplify, observe that since:

\(\cot x = \frac{\cos(2x)}{\sin(2x)}\), it can be related to the identity \(\log(\sin(2x))\) upon integration, leading to the final result:

\(\ln(\sqrt{\sin 2x}) + C\)

This matches with the given options, which confirms the answer: log\(\sqrt{sin\,2x}\)\(+c\)