Question

What is $\mathrm{cosec}x$integration.

Answer 1

Finding the integration of $\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\mathit{x}$:

Multiply and divide the integrand with $\left(\mathrm{cosec}x–\cot x\right)$

$$\begin{gathered} \int \mathrm{cosec}xdx=\int \mathrm{cosec}x\times \frac{\left(\mathrm{cosec}x–\cot x\right)}{\left(\mathrm{cosec}x–\cot x\right)}dx \\ =\int \frac{\left(\cos e{c}^{2}x–\mathrm{cosec}x\cot x\right)}{\left(\cos ecx–cotx\right)}dx...\left(1\right) \end{gathered}$$

Let,

$$\begin{gathered} \mathrm{cosec}x-\cot x=t \\ \Rightarrow \left(-\mathrm{cosec}x\cot x+\cos e{c}^{2}x\right)dx=dt \end{gathered}$$.

Now, substitute the above values in the .$eq.\left(1\right)$:

$$\begin{gathered} \int \mathrm{cosec}xdx=\int \frac{dt}{t} \\ =\log \left|t\right|+c\left[\because \int \frac{dx}{x}=ln\left|x\right|\right] \\ =\log \left|\left(\mathrm{cosec}x-\cot x\right)\right|+c \end{gathered}$$

Thus, the integral of $\mathbf{cosec}\mathbf{}\mathbf{x}\mathbf{}$ is $\mathit{l}\mathit{o}\mathit{g}\mathbf{}\left|\left(cosecx-cotx\right)\right|\mathbf{}\mathbf{+}\mathit{c}$.