Question

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

$\int x{\csc }^2{x}^2\mathrm{d}x$

Short answer

The solution of the given integral is $\int {\mathrm{xcsc}}^2{\mathrm{x}}^2\mathrm{dx}=-\frac{1}{2}{\mathrm{cotx}}^2+C$.

Step-by-step solution

Step 1. Given Information 

Solving the given integrals.

$\int x{\csc }^2{x}^2\mathrm{d}x$

Step 2. Solving the given integral using substitution method. 

Let

$$\begin{gathered} u=x^2 \\ \frac{du}{dx}=2x \\ du=2xdx \\ \frac{1}{2}du=xdx \end{gathered}$$

Step 3. This substitution changes the integral into 

$$\begin{gathered} \int x{\csc }^2{x}^2\mathrm{d}x=\frac{1}{2}\int {\csc }^{2}u\mathrm{du} \\ \int {\mathrm{xcsc}}^2{\mathrm{x}}^2\mathrm{dx}=\frac{1}{2}\left[-\mathrm{cotu}\right]+C \\ \int {\mathrm{xcsc}}^2{\mathrm{x}}^2\mathrm{dx}=\frac{1}{2}\left[-{\mathrm{cotx}}^2\right]+C \\ \int {\mathrm{xcsc}}^2{\mathrm{x}}^2\mathrm{dx}=-\frac{1}{2}{\mathrm{cotx}}^2+C \end{gathered}$$