Q. 66 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}=\frac{1}{2}(\ln {\left(\mathrm{sinx}\right))}^2+\mathrm{C}$.

Step 1. Given Information

Solving the given integrals.

$\int \cot x\ln \left(\sin x\right)\mathrm{d}x$

Step 2. Using the substitution method.

$$\begin{gathered} u=\ln \left(\sin x\right) \\ \frac{du}{dx}=\frac{1}{\sin x}·\cos x \\ \frac{du}{dx}=\frac{\cos x}{\sin x} \\ \frac{du}{dx}=\cot x \\ du=\cot xdx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int \cot x\ln \left(\sin x\right)\mathrm{d}x=\int u\mathrm{du} \\ \int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}=\left[\frac{{\mathrm{u}}^{1+1}}{1+1}\right]+\mathrm{C} \\ \int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}=\frac{{\mathrm{u}}^2}{2}+\mathrm{C} \\ \int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}=\frac{1}{2}{\mathrm{u}}^2+\mathrm{C} \\ \int \mathrm{cotxln}\left(\mathrm{sinx}\right)\mathrm{dx}=\frac{1}{2}(\ln {\left(\mathrm{sinx}\right))}^2+\mathrm{C} \end{gathered}$$