definite integral from 0 to pi/7 of sec(x)tan(x)

The integral you are asking for is:

\[
\int_0^{\frac{\pi}{7}} \sec(x) \tan(x) \, dx
\]

We know that the derivative of \(\sec(x)\) is \(\sec(x) \tan(x)\). Therefore, we can directly integrate as follows:

\[
\int \sec(x) \tan(x) \, dx = \sec(x) + C
\]

Now, we apply the limits from 0 to \(\frac{\pi}{7}\):

\[
\left[\sec(x)\right]_0^{\frac{\pi}{7}} = \sec\left(\frac{\pi}{7}\right) - \sec(0)
\]

Since \(\sec(0) = 1\), the result simplifies to:

\[
\sec\left(\frac{\pi}{7}\right) - 1
\]

This is the exact value of the definite integral:

\[
\boxed{\sec\left(\frac{\pi}{7}\right) - 1}
\]

Would you like further details on how this was solved or explanations on secant and tangent functions? Here are some related questions:

1. What is the geometric interpretation of the secant and tangent functions?
2. How can we apply substitution methods to similar integrals?
3. What are common derivatives involving secant and tangent?
4. How would the integral change if the limits were different?
5. What are some applications of integrals involving trigonometric functions?

**Tip:** When dealing with integrals of trigonometric functions, recognizing derivatives of standard functions can simplify the problem greatly.

Explore the step-by-step solution to the definite integral of sec(x)tan(x) from 0 to pi/7. This calculus problem involves key concepts such as the integral of trigonometric functions and the application of the Fundamental Theorem of Calculus, suitable for students in Grades 11-12.

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