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Chapter 1: Problem 395

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. $$ \int_{1}^{\sqrt{2}} \frac{d x}{|x| \sqrt{x^{2}-1}} $$

Short Answer

Expert verified
The integral evaluates to \( \frac{\pi}{4} \).

Step by step solution

01

Identify the Form of the Integral

The integral \( \int \frac{dx}{|x| \sqrt{x^2 - 1}} \) is of the form that can be transformed using a trigonometric substitution, specifically for the inverse secant function. The standard form for the inverse secant is \( \int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \sec^{-1} \left( \frac{x}{a} \right) + C \). Here, \( a=1 \).
02

Simplify the Integral

Given that the absolute value function \(|x|\) does not affect the domain from \(x=1\) to \(x=\sqrt{2}\), it can be simplified to \(x\) as it is positive in this interval. Thus, the integral becomes \( \int \frac{dx}{x \sqrt{x^2 - 1}} \).
03

Integrate Using the Inverse Secant Formula

Now apply the formula directly: \( \int \frac{dx}{x \sqrt{x^2 - 1}} = \sec^{-1}(x) + C \), where \( C \) is the constant of integration. So, \( F(x) = \sec^{-1}(x) \).
04

Evaluate the Definite Integral

Now, evaluate the definite integral from \(1\) to \(\sqrt{2}\): \[ \int_{1}^{\sqrt{2}} \frac{dx}{x \sqrt{x^2 - 1}} = \left[ \sec^{-1}(x) \right]_{1}^{\sqrt{2}} = \sec^{-1}(\sqrt{2}) - \sec^{-1}(1) \].
05

Compute the Values of the Inverse Secant

Calculate \( \sec^{-1}(\sqrt{2}) \) and \( \sec^{-1}(1) \): - \( \sec^{-1}(\sqrt{2}) = \frac{\pi}{4} \) because \( \sec(\frac{\pi}{4}) = \sqrt{2} \). - \( \sec^{-1}(1) = 0 \) because \( \sec(0) = 1 \).
06

Final Result

Substitute the values back into the expression: \[ \sec^{-1}(\sqrt{2}) - \sec^{-1}(1) = \frac{\pi}{4} - 0 = \frac{\pi}{4} \].

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Key Concepts

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

Trigonometric Substitution
Trigonometric substitution is a powerful technique used to simplify complex integrals by leveraging trigonometric identities. In this particular exercise, trigonometric substitution helps evaluate an integral involving the square root of a quadratic expression. Here are some key points to understand:
  • We are dealing with an integral of the form \( \frac{dx}{x \sqrt{x^2 - 1}} \), which aligns with the standard inverse secant form \( \frac{dx}{x \sqrt{x^2 - a^2}} \) where \( a = 1 \).
  • By recognizing this form, we can use the formula for the inverse secant function, simplifying our operations.
  • The process involves substituting the integral into a familiar structure, allowing for simpler evaluation using inverse trigonometric functions.
This substitution reduces the complexity of the problem, transforming it into a form that can be integrated using standard trigonometric integral formulas.
Inverse Secant Function
The inverse secant function, \( \sec^{-1}(x) \), arises in calculus whenever an integral involves a fraction with a square root of the form \( \frac{dx}{x \sqrt{x^2 - a^2}} \). Understanding this function includes:
  • The inverse secant function is unique because it helps calculate angles given their secant.
  • It is crucial for converting challenging integrals into more manageable forms that utilize well-known trigonometric identities.
  • The standard formula used is \( \sec^{-1}(x) = \arccos(\frac{1}{x}) \), making it useful in determining angles in various contexts.
By applying this formula properly, such as in our exercise between 1 and \( \sqrt{2} \), it's possible to compute definite integrals with greater ease and accuracy, simplifying complex expressions through inverse trigonometry.
Definite Integral Evaluation
Evaluating definite integrals involves calculating the integral of a function over a specific interval, resulting in a real number. In the context of this exercise:
  • The range is from \( x = 1 \) to \( x = \sqrt{2} \), which simplifies due to the properties of the functions involved.
  • The definite integral is calculated by finding the antiderivative, then evaluating it at the upper and lower bounds of the interval.
  • For example, as shown: substitute into \( [ \sec^{-1}(x) ]_1^{\sqrt{2}} \), resulting in \( \sec^{-1}(\sqrt{2}) - \sec^{-1}(1) \).
Specifically, knowing \( \sec^{-1}(\sqrt{2}) = \frac{\pi}{4} \) and \( \sec^{-1}(1) = 0 \), students arrive at the final result of \( \frac{\pi}{4} \). This emphasizes the importance of understanding both the inverse trigonometric functions and the limits of integration in achieving an accurate solution.

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Most popular questions from this chapter

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using \(u\) -substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with \(a\) the left endpoint of the given interval. $$ \text { [T] } \int 3 x^{2} \sqrt{2 x^{3}+1} d x \text { over }[0,1] $$

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