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You have uploaded an image of an integral expression. The expression reads as follows:

\[
\int \sqrt{\tan^2(w)} \cdot \sec(w) \, dw
\]

Let's break this down and simplify the integral:

1. \(\sqrt{\tan^2(w)} = |\tan(w)|\), since the square root of a square is the absolute value of the function.
2. Therefore, the integral becomes:
\[
\int |\tan(w)| \cdot \sec(w) \, dw
\]
Now, let’s consider two cases based on the sign of \(\tan(w)\):

**Case 1:** \( w \in [0, \pi/2] \), where \(\tan(w) \geq 0\). This simplifies the integral to:
\[
\int \tan(w) \cdot \sec(w) \, dw
\]
Using the identity \(\sec^2(w) = 1 + \tan^2(w)\), we can rewrite this as:
\[
\int \sec(w) \cdot \tan(w) \, dw = \sec(w) + C
\]

Thus, the integral simplifies to:
\[
\sec(w) + C
\]

Would you like me to work out the bounds for this definite integral? Or do you have any questions on the process so far?

Here are 5 related questions that you might find useful:
1. How do we handle integrals with absolute value functions?
2. What is the significance of the secant and tangent functions in trigonometry?
3. Can you explain how to integrate trigonometric functions in general?
4. How do we handle definite integrals with piecewise functions?
5. What are other methods to evaluate integrals involving trigonometric identities?

**Tip:** When integrating, always consider simplifying the expression using known identities before diving into complex steps.

Evaluate the integral: \( \int \sqrt{\tan^2(w)} \cdot \sec(w) \, dw \) over the interval [0, 2].

Learn how to evaluate the integral \( \int \sqrt{\tan^2(w)} \cdot \sec(w) \, dw \). This detailed solution covers key calculus concepts, including trigonometric functions, absolute values, and identities. Suitable for undergraduate calculus students.

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