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Magazine Chapter 7: Problem 5
Evaluate the integral. \( \displaystyle \int \frac{\sqrt{x^2 - 1}}{x^4}\ dx \)
Short Answer
Expert verified
The evaluated integral is \( \frac{2\sqrt{x^2-1}}{3x} + \frac{\sqrt{(x^2-1)^3}}{3x^3} + C \).
Step by step solution
01
Recognize the Function Form
The given integral is \( \int \frac{\sqrt{x^2 - 1}}{x^4}\, dx \). Notice that the integral has a square root in the numerator and a polynomial in the denominator. This suggests that substitution might simplify the integration process.
02
Choose a Suitable Substitution
To simplify \( \sqrt{x^2 - 1} \), you can use the trigonometric substitution \( x = \sec(\theta) \). This is because \( \sec^2(\theta) - 1 = \tan^2(\theta) \), leading to \( \sqrt{x^2-1} = \tan(\theta) \). Moreover, \( dx = \sec(\theta)\tan(\theta)\,d\theta \).
03
Substitute and Simplify the Integral
Substitute \( x = \sec(\theta) \):- \( \sqrt{x^2 - 1} = \tan(\theta) \)- \( dx = \sec(\theta)\tan(\theta)\,d\theta \)- The integral becomes \( \int \frac{\tan(\theta) \cdot \sec(\theta) \tan(\theta)}{(\sec(\theta))^4}\,d\theta \)This simplifies to \( \int \frac{\tan^2(\theta)}{\sec^3(\theta)}\,d\theta \). Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), further simplify to \( \int \tan^2(\theta) \cos^3(\theta)\,d\theta \).
04
Use Trigonometric Identity
Use the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) to rewrite the integrand:\[ \int \cos^3(\theta) (\sec^2(\theta) - 1)\,d\theta \] This becomes two separate integrals: \( \int \cos(\theta) - \int \cos^3(\theta)\,d\theta \).
05
Evaluate the Separate Integrals
Evaluate the integrals separately:1. \( \int \cos(\theta)d\theta = \sin(\theta) + C_1 \)2. Use substitution for \( \int \cos^3(\theta)d\theta \). Let \( u = \sin(\theta) \), then \( du = \cos(\theta)d\theta \), so \( \int \cos^2(\theta) \cos(\theta) \right) = \int (1-u^2)\,du = u - \frac{u^3}{3} + C_2 \).Simplify to \( \sin(\theta) - \frac{\sin^3(\theta)}{3} + C_2 \).
06
Combine and Back-Substitute
Combine results from Step 5:\[ \sin(\theta) - (\sin(\theta) - \frac{\sin^3(\theta)}{3}) = \frac{2}{3}\sin(\theta) + \frac{1}{3}\sin^3(\theta) \] Convert back to \( x \): Since \( x = \sec(\theta) \), \( \sin(\theta) = \sqrt{x^2-1}/x \). Thus,\[ \frac{2}{3}\left(\frac{\sqrt{x^2-1}}{x}\right) + \frac{1}{3}\left(\frac{\sqrt{x^2-1}}{x}\right)^3 + C \].
07
Simplify the Expression
Simplify and rationalize the expression:- The integral becomes:\[ \frac{2\sqrt{x^2-1}}{3x} + \frac{\sqrt{(x^2-1)^3}}{3x^3} + C \].This is the evaluated integral with \( C \) as the constant of integration.
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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 is a way of calculating the area between a curve and the x-axis within a fixed interval. Unlike an indefinite integral, which provides a general expression or antiderivative, a definite integral evaluates the exact area over a specified interval.
- It has upper and lower limits, which define the boundaries of integration.
- The result of a definite integral is a singular numerical value, rather than a function plus a constant of integration.
Integration Techniques
Integration techniques are strategies used to evaluate integrals, particularly when straightforward integration isn't possible due to complexity. Various techniques come in handy in different scenarios:
- Substitution: Simplifies integrals by changing variables, making it easier to manage difficult integrands.
- Integration by Parts: Useful when the integrand is a product of two functions. It employs parts of the derivatives and antiderivatives of functions to simplify the integral.
- Partial Fractions: Decomposes a complex rational function into simpler fractions that are easier to integrate.
Substitution Method
The substitution method is a common technique in integration used to tackle complex integrals by transforming them into simpler ones. This method is akin to the reverse of the chain rule used in differentiation.
- Choosing a Suitable Substitution: The aim is to select a trigonometric function that matches the pattern within the integral. For this exercise, recognizing the expression \(\sqrt{x^2-1}\) led to selecting \(x = \sec(\theta)\).
- Substituting in the Integral: Replaced \(x\) with expressions of \(\theta\), turning the integral into an expression involving simpler trigonometric functions.
Trigonometric Identities
Trigonometric identities play a pivotal role in simplifying integrals, especially when the integral involves complex trigonometric expressions. Key trigonometric identities include:
- Pythagorean Identities: These relate the square of sine, cosine, and tangent functions. For example, \(\tan^2(\theta) = \sec^2(\theta) - 1\) is used in this integral.
- Angle Sum and Difference Identities: Important for decomposing and transforming expressions involving angles.
