Problem 17
Question
Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{d x}{\sqrt{x^{2}+2 x+5}}\)
Step-by-Step Solution
Verified Answer
\(\ln \left| \frac{\sqrt{x^2 + 2x + 5} + x + 1}{2} \right| + C\)
1Step 1: Simplify the Radicand (Complete the Square)
The expression under the square root is \(x^2 + 2x + 5\). To complete the square, we express it in the form \((x + a)^2 + b\). We have:\[x^2 + 2x + 5 = (x + 1)^2 + 4\]Here, \((x + 1)^2 = x^2 + 2x + 1\), so we add 4 to complete the expression as: \[(x + 1)^2 + 4 = x^2 + 2x + 5\].
2Step 2: Perform Trigonometric Substitution
Let \( x + 1 = 2 \tan \theta \), then \( dx = 2 \sec^2 \theta \, d\theta \). The substitution simplifies the expression inside the square root:\[ \sqrt{(x+1)^2 + 4} = \sqrt{4\tan^2 \theta + 4} = 2\sec \theta\].This simplifies the integral to:\[ \int \frac{dx}{\sqrt{x^2+2x+5}} = \int \frac{2\sec^2 \theta \, d\theta}{2\sec \theta} = \int \sec \theta \, d\theta\].
3Step 3: Integrate Using Known Integrals
The integral \(\int \sec \theta \, d\theta\) is a standard result, and its solution is:\[ \int \sec \theta \, d\theta = \ln |\sec \theta + \tan \theta| + C\].
4Step 4: Back-Substitution
Recall that \(x + 1 = 2\tan \theta\), so \(\tan \theta = \frac{x + 1}{2}\) and \(\sec \theta = \sqrt{1 + \tan^2 \theta} = \sqrt{\frac{x^2 + 2x + 5}{4}} = \frac{\sqrt{x^2 + 2x + 5}}{2}\). Therefore, the back-substitution is:\[\sec \theta + \tan \theta = \frac{\sqrt{x^2 + 2x + 5} + x + 1}{2}\].Thus, the final result in terms of \(x\) is:\[\ln \left| \frac{\sqrt{x^2 + 2x + 5} + x + 1}{2} \right| + C\].
Key Concepts
Integration TechniquesTrigonometric SubstitutionCompleting the SquareDefinite and Indefinite Integrals
Integration Techniques
To solve integrals, we often need to use a variety of techniques. Some popular methods include substitution, integration by parts, and partial fraction decomposition. Depending on the type of integral, different techniques or combinations of techniques might be used.
One common technique is **trigonometric substitution**, which is particularly useful for integrals involving square roots. When dealing with expressions such as \(\sqrt{x^2 + a^2}\), we can use substitutions like \(x = a\tan\theta\) to make the integral simpler.
Another technique is **completing the square**, which can transform quadratic expressions under a square root into a more manageable form. By altering the expression into a perfect square plus a constant, we simplify the integration process.
By mastering these techniques, you can handle a variety of complex integrals, transforming them into basic forms that are easier to solve.
One common technique is **trigonometric substitution**, which is particularly useful for integrals involving square roots. When dealing with expressions such as \(\sqrt{x^2 + a^2}\), we can use substitutions like \(x = a\tan\theta\) to make the integral simpler.
Another technique is **completing the square**, which can transform quadratic expressions under a square root into a more manageable form. By altering the expression into a perfect square plus a constant, we simplify the integration process.
By mastering these techniques, you can handle a variety of complex integrals, transforming them into basic forms that are easier to solve.
Trigonometric Substitution
When faced with integrals that involve square roots, trigonometric substitution can greatly simplify the problem. This technique uses identities from trigonometry to transform the integral into a more straightforward form.
In our original exercise, we encountered the term \(\sqrt{x^2 + 2x + 5}\). After completing the square in Step 1, this expression became \(\sqrt{(x+1)^2 + 4}\).
For an expression like \((x + a)^2 + b^2\), substituting \(x + a = b\tan\theta\) and using the identity \(1 + \tan^2\theta = \sec^2\theta\) simplifies the integral. The derivative \(dx\) also changes to incorporate \(\sec^2\theta\), aligning with the trigonometric function.
By substituting and simplifying, we transformed our integral into \(\int \sec \theta \, d\theta\), a much easier form.
In our original exercise, we encountered the term \(\sqrt{x^2 + 2x + 5}\). After completing the square in Step 1, this expression became \(\sqrt{(x+1)^2 + 4}\).
For an expression like \((x + a)^2 + b^2\), substituting \(x + a = b\tan\theta\) and using the identity \(1 + \tan^2\theta = \sec^2\theta\) simplifies the integral. The derivative \(dx\) also changes to incorporate \(\sec^2\theta\), aligning with the trigonometric function.
By substituting and simplifying, we transformed our integral into \(\int \sec \theta \, d\theta\), a much easier form.
Completing the Square
Completing the square is a crucial algebraic technique when dealing with polynomial expressions, especially under a square root in integrals. It helps to restructure a quadratic expression so that it's easier to manage.
In the integral \(\int \frac{d x}{\sqrt{x^{2}+2 x+5}}\), the expression under the square root is a quadratic one. Completing the square for \(x^2 + 2x + 5\) involves transforming it to \((x + 1)^2 + 4\).
The process involves:
In the integral \(\int \frac{d x}{\sqrt{x^{2}+2 x+5}}\), the expression under the square root is a quadratic one. Completing the square for \(x^2 + 2x + 5\) involves transforming it to \((x + 1)^2 + 4\).
The process involves:
- Identifying the coefficient of \(x\) (which is 2 in our example),
- Taking half of it (\(2/2 = 1\)),
- Squaring the result to prepare to add and subtract to create a perfect square. In this case, \((x + 1)^2\) was formed.
Definite and Indefinite Integrals
Calculus divides integrals into two main types: definite and indefinite. Understanding these concepts helps when applying integration techniques.
An **indefinite integral** is used to find functions (antiderivatives) whose derivative is the original function. They do not have specific bounds and include an arbitrary constant \(C\), representing the family of all antiderivatives. In our exercise, we worked with an indefinite integral, leading to the solution \(\ln \left| \frac{\sqrt{x^2 + 2x + 5} + x + 1}{2} \right| + C\).
A **definite integral**, on the other hand, computes the net area under a curve between two specific points, leading to a real number as a result. It involves evaluating the antiderivative at these bounds.
Developing a clear understanding of these types helps in choosing and applying the right integration techniques for solving various problems.
An **indefinite integral** is used to find functions (antiderivatives) whose derivative is the original function. They do not have specific bounds and include an arbitrary constant \(C\), representing the family of all antiderivatives. In our exercise, we worked with an indefinite integral, leading to the solution \(\ln \left| \frac{\sqrt{x^2 + 2x + 5} + x + 1}{2} \right| + C\).
A **definite integral**, on the other hand, computes the net area under a curve between two specific points, leading to a real number as a result. It involves evaluating the antiderivative at these bounds.
Developing a clear understanding of these types helps in choosing and applying the right integration techniques for solving various problems.
Other exercises in this chapter
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