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Chapter 6: Problem 21

Evaluate the given improper integral. \(\int_{0}^{\pi} \sec ^{2} x d x\)

Short Answer

Expert verified
The integral evaluates to 0.

Step by step solution

01

Understand the integral and identify substitutions

The given integral is \( \int_{0}^{\pi} \sec^2 x \, dx \). We need to evaluate this improper integral over the interval \([0, \pi]\). Remember that \( \sec^2 x \) is the derivative of \( \tan x \).
02

Set up the indefinite integral

Since the integral of \( \sec^2 x \) is \( \tan x \), we can write the antiderivative as \( \tan x + C \), where \( C \) is the constant of integration.
03

Evaluate the definite integral from 0 to \( \pi \)

Now, evaluate the definite integral using the Fundamental Theorem of Calculus: \[ \int_{0}^{\pi} \sec^2 x \, dx = \left[ \tan x \right]_{0}^{\pi} = \tan(\pi) - \tan(0). \]
04

Substitute the bounds into the antiderivative

Evaluate \( \tan(\pi) \) and \( \tan(0) \). We know that \( \tan(\pi) = 0 \) and \( \tan(0) = 0 \). Therefore, \[ \left[ \tan x \right]_{0}^{\pi} = 0 - 0 = 0. \]
05

Analyze for improper behavior

Although the function \( \sec^2 x \) is undefined at \( x = \frac{\pi}{2} \), the symmetry of the function and its antiderivative results in a cancellation, leading to a finite value of 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
In calculus, the concept of an antiderivative is about finding a function whose derivative gives us back our original function. In simpler terms, if we're given a derivative, the antiderivative is the function we started with before differentiating it. For our integral \( \int \sec^2 x \, dx \), secant squared \( (\sec^2 x) \) is the derivative of the tangent function \( (\tan x) \). This means that \( \tan x \) is the antiderivative of \( \sec^2 x \). Knowing this, it allows us to rewrite indefinite integrals in terms of known functions plus a constant \( C \), which is important because differentiation eliminates constants.
  • The antiderivative of simple trigonometric functions is crucial for solving integrals involving them.
  • It turns the process of integrating from one of raw computation into using known forms.
Definite Integral
A definite integral is an integral with specified limits of integration, such as \( a \) and \( b \). For our exercise, this is the interval from 0 to \( \pi \). Unlike indefinite integrals which have the constant \( C \), definite integrals provide a precise value representing the signed area under the curve. Evaluating the definite integral \( \int_{0}^{\pi} \sec^2 x \, dx \) involves finding the antiderivative \( \tan x \) at the bounds 0 and \( \pi \), then subtracting these values. To solve, we compute:
  • \( \left[ \tan x \right]_{0}^{\pi} = \tan(\pi) - \tan(0) \)
  • Both \( \tan(0) \) and \( \tan(\pi) \) are 0, giving us a final result of 0.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus serves as a bridge between derivatives and integrals, showing how these concepts are interconnected. It can be divided into two main parts: the first part establishes that the process of differentiation can be reversed by integration, leading to the idea of the antiderivative. The second part shows that the definite integral of a function can be computed using its antiderivative. In solving our definite integral \( \int_{0}^{\pi} \sec^2 x \, dx \), this theorem is critical. It tells us that:
  • We can find the integral's result by calculating the antiderivative at the bounds set by the limits (in our case, 0 and \( \pi \)).
  • By computing \( \tan(\pi) - \tan(0) = 0 - 0 \), it confirms that even if the function seems undefined at a point within the interval, the structure of the integral can provide a finite, real result.

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Most popular questions from this chapter

Evaluate the given improper integral. \(\int_{0}^{1} x \ln x d x\)

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the given definite integral converges or diverges. Clearly state what test is being used and what function the integrand is being compared to. \(\int_{0}^{\infty} \frac{\sqrt{x+3}}{\sqrt{x^{3}-x^{2}+x+1}} d x\)

Evaluate the given limit. \(\lim _{x \rightarrow \infty} \frac{\ln \left(x^{2}\right)}{x}\)

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the given definite integral converges or diverges. Clearly state what test is being used and what function the integrand is being compared to. \(\int_{2}^{\infty} \frac{1}{x^{2}+\sin x} d x\)

Evaluate the given limit. \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x}\)
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