Integration of √(1 + cosec(x))

To integrate the expression √(1 + cosec(x)), we can use different techniques such as trigonometric identities and substitution. Let's break down the process into steps for better understanding.

Step 1: Simplify the expression
We start by simplifying the expression inside the square root. The trigonometric identity that relates cosec(x) to sin(x) is:

cosec^2(x) = 1 + cot^2(x)

Rearranging this equation, we get:

cot^2(x) = cosec^2(x) - 1

Now, substitute this into the original expression:

√(1 + cosec(x)) = √(1 + cosec^2(x) - 1) = √(cosec^2(x)) = |cosec(x)|

Note that we used the absolute value sign because cosec(x) can be negative depending on the quadrant of x.

Step 2: Transform the expression
To simplify the absolute value of cosec(x), we can rewrite it using the reciprocal identity:

|cosec(x)| = |1/sin(x)| = 1/|sin(x)|

Step 3: Evaluate the integral
Now, we can proceed with integrating the expression:

∫ √(1 + cosec(x)) dx = ∫ (1/|sin(x)|) dx

To evaluate this integral, we can use a trigonometric substitution. Let's substitute sin(x) = t:

dx = dt/cos(x)

The integral then becomes:

∫ (1/|sin(x)|) dx = ∫ (1/|t|) (dt/cos(x))

Since |t| is always positive, we can remove the absolute value sign:

∫ (1/|sin(x)|) dx = ∫ (1/t) (dt/cos(x)) = ∫ (dt/cos(x))

Step 4: Simplify the integral
The integral ∫ (dt/cos(x)) can be evaluated as:

∫ (dt/cos(x)) = ∫ sec(x) dx = ln|sec(x) + tan(x)| + C

where C is the constant of integration.

Step 5: Final result
Combining all the steps, we have:

∫ √(1 + cosec(x)) dx = ln|sec(x) + tan(x)| + C

where C is the constant of integration.