Question

If $\int \mathrm{cosec}xdx=f\left(x\right)+\mathrm{constant}$, then $f\left(x\right)=$

• A. $\tan \frac{x}{2}$
• B. $\log \left|\tan \left(\frac{x}{2}\right)\right|$
• C. $\log \left|\sin x\right|$
• D. $\log \left|\cos x\right|$

Answer 1

The correct option is B

$\log \left|\tan \left(\frac{x}{2}\right)\right|$

Explanation for the correct option:

Find the function $f\left(x\right)$.

It is known that the $\int \mathrm{cosec}xdx=-\log \left|\cot \left(\frac{x}{2}\right)\right|+\mathrm{constant}$.

So the function $f\left(x\right)$ is $-\log \left|\cot \left(\frac{x}{2}\right)\right|$ and it can be written as:

$\begin{array}{rcl}f\left(x\right)& =& -\log \left|\cot \left(\frac{x}{2}\right)\right|\\ & =& \log \left|{\left(\cot \left(\frac{x}{2}\right)\right)}^{-1}\right|\\ & =& \log \left|\tan \left(\frac{x}{2}\right)\right|\left[\frac{\mathbf{1}}{\mathbf{cot}\mathbf{x}}\mathbf{=}\mathbf{tan}\mathit{x}\right]\end{array}$

Thus the function $f\left(x\right)$ is $\log \left|\tan \left(\frac{x}{2}\right)\right|$.

Hence, the correct option is B.