Question

Find the integral of $\int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx$.

Answer 1

Find the integral of the given function.

Given integral: $\int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx.$

Let, $\log \left(\tan \frac{x}{2}\right)=t$

By differentiating both sides we get,

$$\begin{gathered} \frac{{\displaystyle \frac{1}{2}}se{c}^2\frac{x}{2}}{\tan \frac{x}{2}}dx=dt \\ \Rightarrow \frac{\frac{1}{2}se{c}^2\frac{x}{2}}{\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}}dx=dt \\ \Rightarrow \frac{\frac{1}{2}se{c}^2\frac{x}{2}}{\sin \frac{x}{2}sec\frac{x}{2}}dx=dt \\ \Rightarrow \frac{sec\frac{x}{2}}{2\sin \frac{x}{2}}dx=dt \\ \Rightarrow \frac{1}{2\sin \frac{x}{2}\cos \frac{x}{2}}dx=dt \\ \Rightarrow \frac{1}{\sin x}dx=dt\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{2}\mathbf{\theta }\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{]} \\ \Rightarrow \cos ecxdx=dt \end{gathered}$$

So, $\int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx=\int \frac{dt}{t}\left[\because \log \left(tan\frac{x}{2}\right)=t\mathrm{and}cosecxdx=dt\right]$

$=\log \left|t\right|+C$, where $C$ is the integration constant.

$\mathbf{}=\log \left|\log (tan\frac{x}{2})\right|+C\mathbf{[}\mathbf{\because }\mathbf{log}\mathbf{(}\mathbf{tan}\mathbf{}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{t}\mathbf{]}$

$\therefore \int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx=\log \left|\log (tan\frac{x}{2})\right|+C$

Hence, the integral of $\int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx$ is $\log \left|\log (tan\frac{x}{2})\right|+C$, where $C$ is the integration constant.