i am getting no idea how to fit it in the form of antiderivative of cosecinverse x pls give me directions to proceed with

To find the antiderivative of the cosecant inverse function, denoted as \( \text{csc}^{-1}(x) \), we can use a systematic approach. The antiderivative, or integral, of a function is essentially the reverse process of differentiation. For \( \text{csc}^{-1}(x) \), we can utilize integration techniques that involve trigonometric identities and substitutions. Let’s break this down step by step.

Understanding the Function

The cosecant inverse function, \( \text{csc}^{-1}(x) \), is defined for \( |x| \geq 1 \). Its derivative is given by:

• \( \frac{d}{dx} \text{csc}^{-1}(x) = -\frac{1}{|x| \sqrt{x^2 - 1}} \)

This derivative will be useful in our integration process. To find the antiderivative, we can use integration by parts or a direct integration approach.

Using Integration by Parts

Integration by parts is based on the formula:

• \( \int u \, dv = uv - \int v \, du \)

We can choose \( u = \text{csc}^{-1}(x) \) and \( dv = dx \). Then, we need to find \( du \) and \( v \):

• From the derivative, we have \( du = -\frac{1}{|x| \sqrt{x^2 - 1}} \, dx \)
• And \( v = x \)

Setting Up the Integral

Now, substituting into the integration by parts formula gives us:

• \( \int \text{csc}^{-1}(x) \, dx = x \cdot \text{csc}^{-1}(x) - \int x \left(-\frac{1}{|x| \sqrt{x^2 - 1}}\right) \, dx \)

This simplifies to:

• \( \int \text{csc}^{-1}(x) \, dx = x \cdot \text{csc}^{-1}(x) + \int \frac{x}{|x| \sqrt{x^2 - 1}} \, dx \)

Evaluating the Remaining Integral

Next, we need to evaluate the integral \( \int \frac{x}{|x| \sqrt{x^2 - 1}} \, dx \). For \( x \geq 1 \), this simplifies to:

• \( \int \frac{1}{\sqrt{x^2 - 1}} \, dx \)

This integral can be solved using the substitution \( x = \sec(\theta) \), which leads to:

• \( dx = \sec(\theta) \tan(\theta) \, d\theta \)
• Thus, \( \int \frac{1}{\sqrt{x^2 - 1}} \, dx = \ln |x + \sqrt{x^2 - 1}| + C \)

Final Result

Putting everything together, we have:

• \( \int \text{csc}^{-1}(x) \, dx = x \cdot \text{csc}^{-1}(x) + \ln |x + \sqrt{x^2 - 1}| + C \)

In summary, the antiderivative of \( \text{csc}^{-1}(x) \) is expressed as:

• \( \int \text{csc}^{-1}(x) \, dx = x \cdot \text{csc}^{-1}(x) + \ln |x + \sqrt{x^2 - 1}| + C \)

By following these steps, you can systematically approach the integration of the cosecant inverse function. If you have any further questions or need clarification on any part of the process, feel free to ask!