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Magazine Chapter 8: Problem 32
Evaluate the following integrals. $$\int_{0}^{2} \frac{x}{x^{2}+4 x+8} d x$$
Short Answer
Expert verified
Question: Evaluate the definite integral $\int_{0}^{2} \frac{x}{x^{2}+4 x+8} d x$.
Answer: The definite integral evaluates to $-\frac{\pi}{8}$.
Step by step solution
01
Partial Fraction Decomposition
To perform partial fraction decomposition, complete the square of the denominator, which will help to simplify the integration later on.
The given integral is:
$$\int_{0}^{2} \frac{x}{x^{2}+4 x+8} d x$$
Completing the square of the denominator, we have:
$$x^2+4x+8=(x+2)^{2}+4$$
Now the integral becomes:
$$\int_{0}^{2} \frac{x}{(x+2)^{2}+4} d x$$
02
Apply substitution
Let \(u = x+2\). Then, \(x = u-2\). Now, the integral becomes:
$$\int_{-2}^{0} \frac{u-2}{u^{2}+4} d u$$
03
Integration
Now, we are going to perform the integration. The integral can be separated as:
$$\int_{-2}^{0} \frac{u}{u^2 + 4}\,d u - \int_{-2}^{0} \frac{2}{u^2+4}\,d u$$
To solve the first integral, consider the substitution \(u = 2\tan{v}\). Equivalently, \(d u = 2\sec^2{v}\,d v\). Then, we get
$$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{2\tan{v}\cdot 2\sec^2{v}}{(2\tan{v})^2 + 4}\,d v = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\sin{v}\cdot \cos{v}}{\cos^2{v}}\, d v = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin{v}\,d v = \left[-\cos{v}\right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = -(\cos{\frac{\pi}{4}}-\cos{-\frac{\pi}{4}}) = 0$$
Now, we will find the second integral.
04
Integration of second part
To solve the second integral, we notice that it is a standard integral of the form \(\int \frac{1}{u^2+4}\,d u\). Observe that \(\int\frac{1}{u^2+4}\,d u = \int\frac{1}{u^2+2^2}\,d u = \frac{1}{2}\int\frac{1}{(\frac{u}{2})^2+1}\,d u = \frac{1}{2}\arctan{\frac{u}{2}}+C\).
Now, we will evaluate the definite integral:
$$\frac{1}{2}\left(\arctan{\frac{0}{2}}-\arctan{\frac{-2}{2}}\right) = \frac{1}{2}\left(0-\arctan{-1}\right) = \frac{1}{2}\left(\frac{\pi}{4}\right)$$
05
Combine the results
Now, combining the results of both integrals, we get the answer to the given integral:
$$\int_{0}^{2} \frac{x}{x^{2}+4 x+8} d x = \left(0\right)-\left(\frac{1}{2}\left(\frac{\pi}{4}\right)\right) = -\frac{\pi}{8}$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used in calculus to break down complex rational expressions into simpler fractions that can be integrated more easily. It's particularly useful when dealing with rational functions, where polynomials are in the numerator and the denominator.
The process often involves rewriting the denominator in a way that it can be expressed as a product of linear factors or irreducible quadratic factors. For instance, in our exercise, we complete the square for the denominator to transform it into a sum of a square and a constant. Once in this form, partial fractions can then be applied to separate the complex fraction into simpler fractions, making the subsequent integration more manageable.
The process often involves rewriting the denominator in a way that it can be expressed as a product of linear factors or irreducible quadratic factors. For instance, in our exercise, we complete the square for the denominator to transform it into a sum of a square and a constant. Once in this form, partial fractions can then be applied to separate the complex fraction into simpler fractions, making the subsequent integration more manageable.
Integration Techniques
Integration techniques are an assortment of tools used to calculate the integral, or antiderivative, of a function. These techniques, which include partial fraction decomposition, u-substitution, integration by parts, and trigonometric substitution, allow us to tackle a wide range of integrals.
In this exercise, we employ partial fraction decomposition followed by u-substitution and trigonometric substitution. This multi-step approach ensures that we can integrate functions that at first glance may seem too complicated. By mastering these methods, you can solve most integrals encountered in calculus.
In this exercise, we employ partial fraction decomposition followed by u-substitution and trigonometric substitution. This multi-step approach ensures that we can integrate functions that at first glance may seem too complicated. By mastering these methods, you can solve most integrals encountered in calculus.
U-Substitution
U-substitution is a basic method for finding integrals that is inspired by the chain rule for differentiation. When a given integral contains a function and its derivative, u-substitution involves substituting a part of the integral (typically the inner function of a composition) with a new variable, 'u'.
Let's consider a practical example from the exercise. We substitute the expression 'x+2鈥� with 'u', which simplifies the integral significantly. It is imperative to adjust the limits of integration when working with definite integrals. After substitution, we integrate with respect to the new variable and then substitute back to the original variable once we have found the antiderivative.
Let's consider a practical example from the exercise. We substitute the expression 'x+2鈥� with 'u', which simplifies the integral significantly. It is imperative to adjust the limits of integration when working with definite integrals. After substitution, we integrate with respect to the new variable and then substitute back to the original variable once we have found the antiderivative.
Trigonometric Substitution
Trigonometric substitution is a clever trick used when an integrand involves square roots of quadratic expressions or when the integrand has a quadratic expression that can be turned into a square. It involves substituting parts of the expression with trigonometric identities, thus simplifying the integral.
In our case, the substitution \( u = 2\tan{v} \) is used, which converts the integral into a trigonometric function. This particular substitution was chosen because it results in a direct application of the Pythagorean identity, which simplifies the integral further. The process involves evaluating the trigonometric integral and then using the inverse trigonometric function to express the final answer.
In our case, the substitution \( u = 2\tan{v} \) is used, which converts the integral into a trigonometric function. This particular substitution was chosen because it results in a direct application of the Pythagorean identity, which simplifies the integral further. The process involves evaluating the trigonometric integral and then using the inverse trigonometric function to express the final answer.
