Q. 14 - Calculus

Question

Verify that$\int \cot xdx=\ln \left(\sin x\right)+C$. (Do not try to solve the integral from scratch.) 

Step-by-Step Solution

Answer

It is verified that $\int \cot xdx=\ln \left(\sin x\right)+C.$

Step 1. Given information.

The given integral is $\int \cot xdx=\ln \left(\sin x\right)+C.$

Step 2. Verification.

We can write the differentiation as,

$$\begin{gathered} \frac{d}{dx}\ln \left(\sin x\right) \\ =\frac{1}{\sin x}\left(\cos x\right) \end{gathered}$$

$$\begin{gathered} =\frac{\cos x}{\sin x} \\ =cot x \end{gathered}$$

Hence, $\ln \left(\sin x\right)\text{ is an anti-derivative of }\cot x\text{. }$

Therefore,

$\int \cot xdx=\ln \left(\sin x\right)+C$