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Magazine Chapter 6: Problem 118
Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sec ^{2} x d x $$
Short Answer
Expert verified
The value of the definite integral is 1.
Step by step solution
01
Identify the Integral to Solve
The given integral is \( \int_{0}^{\pi / 4} \sec ^{2} x \, dx \). This is a definite integral, so we will evaluate it over the interval from \( x=0 \) to \( x=\pi / 4 \).
02
Recall the Antiderivative of \( \sec^2 x \)
The antiderivative of \( \sec^2 x \) is \( \tan x \), because differentiating \( \tan x \) gives \( \sec^2 x \).
03
Apply the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus states that to evaluate \( \int_{a}^{b} f(x) \, dx \), we find an antiderivative \( F(x) \) such that \( F'(x) = f(x) \), and then compute \( F(b) - F(a) \). Here, the antiderivative is \( \tan x \), so we evaluate \( \tan \left( \frac{\pi}{4} \right) - \tan(0) \).
04
Evaluate \( \tan \left( \frac{\pi}{4} \right) \) and \( \tan(0) \)
Now we calculate the values of the tangent function at the upper and lower limits. \( \tan \left( \frac{\pi}{4} \right) = 1 \) because the tangent of \( \frac{\pi}{4} \) (or 45 degrees) is 1. Also, \( \tan(0) = 0 \) since the tangent of 0 is 0.
05
Substitute and Simplify
Substitute these values into the expression from the Fundamental Theorem of Calculus: \( \tan \left( \frac{\pi}{4} \right) - \tan(0) = 1 - 0 = 1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
A definite integral represents the accumulation of a function's values over a specific interval on the x-axis. In general, for a function \( f(x) \) over the interval \([a, b]\), the definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \). This integral sums up the areas under the curve defined by \( f(x) \), considering both the magnitude and direction.
- The limits of integration, \( a \) and \( b \), define the interval over which we calculate the integral.
- In our example problem, \( \int_{0}^{\pi / 4} \sec ^{2} x \, dx \), the limits are 0 and \( \pi / 4 \).
- The definite integral provides a numerical value which represents the net area between the curve and the x-axis over the specified interval.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation with integration, establishing a powerful tool in calculus. It provides a way to evaluate definite integrals when we know an antiderivative of the function.
- The theorem has two main parts. The first part states that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).
- This means we can evaluate the integral by finding an antiderivative \( F(x) \) of \( f(x) \), and then subtracting \( F(a) \) from \( F(b) \).
- In our problem, since \( \sec^2 x \)'s antiderivative is \( \tan x \), we compute \( \tan \left( \frac{\pi}{4} \right) - \tan(0) \) to evaluate the integral.
Integration Techniques
Integration techniques are strategies used to find antiderivatives or evaluate integrals. They include basic methods and advanced strategies to handle complex functions.
- Recognizing Basic Antiderivatives: Some functions have known antiderivatives. For example, the derivative of \( \tan x \) is \( \sec^2 x \); therefore, the antiderivative of \( \sec^2 x \) is \( \tan x \).
- U-Substitution: This technique simplifies the integration process by transforming the integral into a simpler form. It's particularly useful when dealing with composite functions.
- Partial Fraction Decomposition: Used for rational functions to break them into simpler fractions for easier integration.
- Integration by Parts: Based on the product rule for differentiation, this technique is useful when integrating the product of two functions.
