Math Problem Statement

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

Solution

The integral you are asking for is:

0π7sec(x)tan(x)dx\int_0^{\frac{\pi}{7}} \sec(x) \tan(x) \, dx

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

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

Now, we apply the limits from 0 to π7\frac{\pi}{7}:

[sec(x)]0π7=sec(π7)sec(0)\left[\sec(x)\right]_0^{\frac{\pi}{7}} = \sec\left(\frac{\pi}{7}\right) - \sec(0)

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

sec(π7)1\sec\left(\frac{\pi}{7}\right) - 1

This is the exact value of the definite integral:

sec(π7)1\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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Definite Integral
Trigonometric Functions

Formulas

Integral of sec(x)tan(x) dx = sec(x) + C
Fundamental Theorem of Calculus

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12