integration of 1/sinx+cosx+secx+tanx+cosecx+cotx with respect to x is

To tackle the integration of the expression \( \frac{1}{\sin x} + \cos x + \sec x + \tan x + \csc x + \cot x \) with respect to \( x \), we can break it down into manageable parts. Each term can be integrated separately, and then we can combine the results. Let's go through this step by step.

Breaking Down the Expression

The expression consists of several trigonometric functions. Here’s a quick recap of what each term represents:

• \(\frac{1}{\sin x}\) is the cosecant function, \(\csc x\).
• \(\cos x\) is the cosine function.
• \(\sec x\) is the secant function, which is \(\frac{1}{\cos x}\).
• \(\tan x\) is the tangent function, which is \(\frac{\sin x}{\cos x}\).
• \(\csc x\) is the cosecant function, which is \(\frac{1}{\sin x}\).
• \(\cot x\) is the cotangent function, which is \(\frac{\cos x}{\sin x}\).

Integrating Each Term

Now, let’s integrate each term one by one:

1. Integrating \(\csc x\)

The integral of \(\csc x\) is:

\[ \int \csc x \, dx = \ln |\csc x - \cot x| + C \]

2. Integrating \(\cos x\)

The integral of \(\cos x\) is straightforward:

\[ \int \cos x \, dx = \sin x + C \]

3. Integrating \(\sec x\)

The integral of \(\sec x\) is:

\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]

4. Integrating \(\tan x\)

The integral of \(\tan x\) is:

\[ \int \tan x \, dx = -\ln |\cos x| + C \]

5. Integrating \(\csc x\) (again)

As mentioned earlier, the integral of \(\csc x\) is:

\[ \int \csc x \, dx = \ln |\csc x - \cot x| + C \]

6. Integrating \(\cot x\)

The integral of \(\cot x\) is:

\[ \int \cot x \, dx = \ln |\sin x| + C \]

Combining the Results

Now that we have the integrals of each term, we can combine them:

\[ \int \left( \frac{1}{\sin x} + \cos x + \sec x + \tan x + \csc x + \cot x \right) dx = \ln |\csc x - \cot x| + \sin x + \ln |\sec x + \tan x| - \ln |\cos x| + \ln |\csc x - \cot x| + \ln |\sin x| + C \]

Final Expression

After combining like terms and simplifying, the final result can be expressed as:

\[ \int \left( \frac{1}{\sin x} + \cos x + \sec x + \tan x + \csc x + \cot x \right) dx = \sin x + \ln |\sec x + \tan x| + \ln |\sin x| + C \]

This gives you a comprehensive view of how to integrate the given expression. Each step is crucial for understanding how the components work together in calculus. If you have any further questions or need clarification on any part, feel free to ask!