Question

Integrate $\int cotx\cos xdx$.

Answer 1

Solve the given integral

Given: $\int cotx\cos xdx$

We know that,

$cotx=\frac{\cos x}{\sin x}$

and,

$$\begin{gathered} {\sin }^{2}x+{\cos }^{2}x=1 \\ \Rightarrow {\cos }^{2}x=1-{\sin }^{2}x \end{gathered}$$

Now, $\int cotx\cos xdx$ can be written as,

$=\int \frac{\cos x}{\sin x}\cos xdx$

$$\begin{gathered} =\int \frac{{\cos }^{2}x}{\sin x}dx \\ =\int \frac{1-{\sin }^{2}x}{\sin x}dx \\ =\int \frac{dx}{\sin x}-\int \sin xdx \\ =\int \cos ecxdx+\cos x \\ =-\ln \left(\left|\cos ecx-cotx\right|\right)+\cos x+C \end{gathered}$$

Hence, the Integral of $\int cotx\cos xdx$ is $-\ln \left(\left|\cos ecx-cotx\right|\right)+\cos x+C$, where $C$ is an arbitrary constant.