https://file.aiquickdraw.com/mathbot/file/17270613884784ru6zp9d.jpg

The integral displayed in the image is:

\[
\int \left( \csc x - \tan x \right)^2 dx
\]

Let's break this down and solve the integral step by step.

### Step 1: Expand the integrand
We expand \((\csc x - \tan x)^2\) to get:

\[
\left( \csc x - \tan x \right)^2 = \csc^2 x - 2 \csc x \tan x + \tan^2 x
\]

### Step 2: Integrating term by term
We now integrate each term separately:

\[
\int \csc^2 x \, dx - 2 \int \csc x \tan x \, dx + \int \tan^2 x \, dx
\]

- The integral of \(\csc^2 x\) is \(-\cot x\).
- The integral of \(\csc x \tan x\) is \(-\csc x\).
- The integral of \(\tan^2 x\) can be written using the identity \(\tan^2 x = \sec^2 x - 1\), so the integral becomes:
  \[
  \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \tan x - x
  \]

### Step 3: Combining results
We now put everything together:

\[
-\cot x + 2 \csc x + \tan x - x + C
\]

Thus, the solution to the integral is:

\[
-\cot x + 2 \csc x + \tan x - x + C
\]

Let me know if you'd like further details or clarification on any of the steps!

### Related Questions:
1. What is the integral of \(\sec^2 x\)?
2. How does the Pythagorean identity \(\tan^2 x = \sec^2 x - 1\) help simplify integrals?
3. How do you differentiate trigonometric functions like \(\csc x\) and \(\tan x\)?
4. What is the geometrical interpretation of trigonometric integrals?
5. Can this method of expansion be applied to other types of expressions?

### Tip:
Whenever you encounter a squared term in an integrand, expanding it and integrating term by term often simplifies the process significantly!

What is the integral of (csc x - tan x)^2?

Learn how to compute the integral of (csc x - tan x)^2 using trigonometric identities and integration techniques. This problem involves expanding the integrand and solving term by term, covering key concepts such as the integral of csc^2 x, the identity tan^2 x = sec^2 x - 1, and more.

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