Evaluate the following integral. r/4 1 (s secx- cos cos sx)²dx 0 x/4 ļ (secx- cos x)²dx= 5 3 4 8 0 (Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Evaluate the following integral. r/4 1 (s secx- cos cos sx)²dx 0 x/4 ļ (secx- cos x)²dx= 5 3 4 8 0 (Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Evaluate the following integral.
\[
\int_{0}^{\pi / 4} \left( \sec x - \cos x \right)^2 dx
\]
**Solution:**
\[
\int_{0}^{\pi / 4} \left( \sec (x) - \cos(x) \right)^2 dx = \frac{5}{4} - \frac{3\pi}{8}
\]
**Instructions**: (Type an exact answer. Use parentheses to clearly denote the argument of each function.)
**Explanation:**
To solve this integral, we need to evaluate the expression within the given limits. Generally, follow these steps:
1. Expand the integrand \(\left( \sec (x) - \cos(x) \right)^2\) to make integration simpler.
2. Integrate the expanded expression term by term over the interval \([0, \frac{\pi}{4}]\).
3. Apply the limits of integration to find the exact numerical answer.
4. Simplify the final expression to achieve the solution in its exact form.
In this problem, detailed steps should lead you to the final answer, which combines standard functions and constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F511d779c-bf10-4315-98b1-4f0038c55113%2F1464bede-cfd7-45d3-a77b-a24906633923%2Fmdvecj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the following integral.
\[
\int_{0}^{\pi / 4} \left( \sec x - \cos x \right)^2 dx
\]
**Solution:**
\[
\int_{0}^{\pi / 4} \left( \sec (x) - \cos(x) \right)^2 dx = \frac{5}{4} - \frac{3\pi}{8}
\]
**Instructions**: (Type an exact answer. Use parentheses to clearly denote the argument of each function.)
**Explanation:**
To solve this integral, we need to evaluate the expression within the given limits. Generally, follow these steps:
1. Expand the integrand \(\left( \sec (x) - \cos(x) \right)^2\) to make integration simpler.
2. Integrate the expanded expression term by term over the interval \([0, \frac{\pi}{4}]\).
3. Apply the limits of integration to find the exact numerical answer.
4. Simplify the final expression to achieve the solution in its exact form.
In this problem, detailed steps should lead you to the final answer, which combines standard functions and constants.
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