Integration of 1/sin x + cos x + tan x + cot x dx.All are in denominator and dx is in numerator

To solve the integral of the expression \( \frac{1}{\sin x + \cos x + \tan x + \cot x} \, dx \), we first need to simplify the integrand. This expression combines several trigonometric functions, and simplifying it will make the integration process more manageable.

Breaking Down the Expression

Let's rewrite the trigonometric functions in terms of sine and cosine:

• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\cot x = \frac{\cos x}{\sin x}\)

Substituting these into the integral gives us:

\( \sin x + \cos x + \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \)

Finding a Common Denominator

To combine the terms, we can find a common denominator, which is \(\sin x \cos x\). Rewriting the expression, we have:

\[ \sin x + \cos x + \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \]

Since \(\sin^2 x + \cos^2 x = 1\), we can simplify this further:

\[ \sin x + \cos x + \frac{1}{\sin x \cos x} \]

Combining Terms

Now, we can express the entire denominator as:

\[ \sin x + \cos x + \frac{1}{\sin x \cos x} \]

This expression is still complex, so let's denote \( u = \sin x + \cos x \). The derivative of \( u \) with respect to \( x \) is:

\[ \frac{du}{dx} = \cos x - \sin x \]

Substituting Back into the Integral

Now, we can express the integral in terms of \( u \). However, we need to express \( dx \) in terms of \( du \). Rearranging gives us:

\[ dx = \frac{du}{\cos x - \sin x} \]

Substituting this into our integral, we can rewrite it as:

\[ \int \frac{1}{u + \frac{1}{\sin x \cos x}} \cdot \frac{du}{\cos x - \sin x} \]

Final Steps

This integral can be quite complex, and solving it may require additional techniques such as partial fractions or numerical methods, depending on the specific values of \( x \) or the limits of integration. However, the key takeaway here is that simplifying the integrand and using substitutions can significantly ease the process of integration.

In summary, while the integral of \( \frac{1}{\sin x + \cos x + \tan x + \cot x} \, dx \) is intricate, breaking it down into simpler components and using substitutions can help in finding a solution. If you have further questions or need clarification on any specific step, feel free to ask!