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Magazine Chapter 7: Problem 17
In Problems \(17-26,\) use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. $$ \int \frac{d x}{\sqrt{x^{2}+2 x+5}} $$
Short Answer
Expert verified
The integral evaluates to \(\ln\left|\frac{\sqrt{x^2 + 2x + 5}}{2} + \frac{x+1}{2}\right| + C\).
Step by step solution
01
Identify and Rewrite the Quadratic Expression
First, identify the quadratic expression under the square root: \(x^{2} + 2x + 5\). To complete the square, rewrite it by forming a perfect square trinomial.
02
Completing the Square
The expression \(x^{2} + 2x + 5\) can be rewritten by completing the square. Start by taking half of the linear coefficient (2): \(\frac{2}{2} = 1\). Square it to get \(1^2 = 1\). Add and subtract 1 inside the expression: \(x^{2} + 2x + 1 - 1 + 5 = (x + 1)^2 + 4\).
03
Substitute the Quadratic Expression
Substitute the completed square back into the integral: \(\int \frac{dx}{\sqrt{(x+1)^2 + 4}}\).
04
Trigonometric Substitution
Use the trigonometric substitution \(x + 1 = 2\tan(\theta)\), which implies \(dx = 2\sec^2(\theta)\,d\theta\). The integral becomes \(\int \frac{2\sec^2(\theta)\, d\theta}{\sqrt{4\sec^2(\theta)}}\).
05
Simplify the Expression
Simplify \(\sqrt{4\sec^2(\theta)} = 2\sec(\theta)\). The integral simplifies to \(\int \sec(\theta)\, d\theta\).
06
Integrate using Identity
The integral \(\int \sec(\theta)\, d\theta = \ln|\sec(\theta) + \tan(\theta)| + C\), where \(C\) is the constant of integration.
07
Revert to the Original Variable
Revert back to the variable \(x\). Recall the substitution \(x + 1 = 2\tan(\theta)\), which means \(\tan(\theta) = \frac{x + 1}{2}\). Substituting back, we get \(\sec(\theta) = \frac{\sqrt{x^2 + 2x + 5}}{2}\) and \(\tan(\theta) = \frac{x+1}{2}\). Thus, the solution becomes \(\ln\left|\frac{\sqrt{x^2 + 2x + 5}}{2} + \frac{x+1}{2}\right| + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
When you encounter a quadratic expression like \(x^2 + 2x + 5\) within an integral, one helpful technique is completing the square. This method transforms the quadratic into a form that's easier to integrate.
For any quadratic expression \(ax^2 + bx + c\), the goal is to express it as \((x-h)^2 + k\). This involves rearranging and adding/subtracting necessary terms.
For any quadratic expression \(ax^2 + bx + c\), the goal is to express it as \((x-h)^2 + k\). This involves rearranging and adding/subtracting necessary terms.
- Start by focusing on the \(x\) terms. For the expression \(x^2 + 2x + 5\), take half of the \(x\) coefficient (which is 2), then square it. So, \((\frac{2}{2})^2 = 1\).
- Add and subtract this value to balance the equation, yielding \((x+1)^2 + 4\).
Trigonometric Substitution
Once the square is completed, trigonometric substitution aids in integrating expressions involving square roots.
If you have an expression like \((x+1)^2 + 4\) under a square root, consider substituting \(x + 1 = 2\tan(\theta)\). This particular substitution aligns with the identity \(1 + \tan^2(\theta) = \sec^2(\theta)\).
If you have an expression like \((x+1)^2 + 4\) under a square root, consider substituting \(x + 1 = 2\tan(\theta)\). This particular substitution aligns with the identity \(1 + \tan^2(\theta) = \sec^2(\theta)\).
- Calculate \(dx\) in terms of \(d\theta\). For \(x + 1 = 2\tan(\theta)\), we find that \(dx = 2\sec^2(\theta)\,d\theta\).
- Replace all \(x\) terms in the integral with trigonometric terms, simplifying the square root to \(2\sec(\theta)\).
Definite Integration
The process of definite integration calculates the accumulation of quantities between a specific interval.
For definite integration, the limits of integration are specified, often applied to compute area under a curve. Unlike indefinite integration which results in a general formula including a constant, definite integration provides a precise numerical result.
For definite integration, the limits of integration are specified, often applied to compute area under a curve. Unlike indefinite integration which results in a general formula including a constant, definite integration provides a precise numerical result.
- Convert the entire function into a form that's simpler to integrate, such as completing the square followed by substitutions.
- Apply trigonometric identities and simplifications efficiently.
- Ensure to adjust the limits of integration if utilizing substitution techniques, considering the changes in variables.
Indefinite Integration
Indefinite integration involves finding the antiderivative of a function, essentially reversing differentiation.
The solution involves determining a general form of function whose derivative matches the original, plus a constant of integration \(C\). For the given integral, applying steps of completing the square and substitution leads to \(\ln|\sec(\theta) + \tan(\theta)| + C\).
The solution involves determining a general form of function whose derivative matches the original, plus a constant of integration \(C\). For the given integral, applying steps of completing the square and substitution leads to \(\ln|\sec(\theta) + \tan(\theta)| + C\).
- This result is then transformed back to the original variable through inverse trigonometric relations.
- This step calls for clear substitution back to \(x\), reintroducing it into \(\sec(\theta)\) and \(\tan(\theta)\) utilizing geometric insights and known identities.
