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Magazine Chapter 7: Problem 45
Evaluate the integral. $$ \int \frac{d x}{2 x^{2}+4 x+7} $$
Short Answer
Expert verified
\( \int \frac{d x}{2 x^{2}+4 x+7} = \frac{\sqrt{\frac{5}{2}}}{5} \tan^{-1}\left(\frac{x+1}{\sqrt{\frac{5}{2}}}\right) + C \)
Step by step solution
01
Complete the Square
To evaluate the integral \( \int \frac{d x}{2 x^{2}+4 x+7} \), first, we need to simplify the quadratic expression in the denominator by completing the square. Start by factoring out the constant 2:\[ 2x^2 + 4x + 7 = 2(x^2 + 2x) + 7 \]Now, complete the square for \(x^2 + 2x\):\[ x^2 + 2x = (x+1)^2 - 1 \]Then, replace back in the expression:\[ 2((x+1)^2 - 1) + 7 = 2(x+1)^2 - 2 + 7 = 2(x+1)^2 + 5 \]
02
Rewrite the Integral
Now, substitute the completed square expression back into the integral:\[ \int \frac{d x}{2(x+1)^2 + 5} \]
03
Simplify the Integral using Substitution
Substitute \( u = x + 1 \), which implies \( du = dx \). The integral becomes:\[ \int \frac{du}{2u^2 + 5} \]
04
Use Trigonometric Substitution
To evaluate \( \int \frac{du}{2u^2 + 5} \), use the substitution \( u = \sqrt{\frac{5}{2}} \tan(\theta) \). Thus, \( du = \sqrt{\frac{5}{2}} \sec^2(\theta) d\theta \). Then the integral becomes:\[ \int \frac{\sqrt{\frac{5}{2}} \sec^2(\theta) d\theta}{2 \left(\frac{5}{2}\right) \tan^2(\theta) + 5} \]Simplifying the denominator simplifies to:\[ 5\sec^2(\theta) \]
05
Evaluate the Trigonometric Integral
Now evaluate the integral:\[ \int \frac{\sqrt{\frac{5}{2}} \sec^2(\theta) d\theta}{5 \sec^2(\theta)} = \int \frac{\sqrt{\frac{5}{2}}}{5} d\theta \]This integrates to:\[ \frac{\sqrt{\frac{5}{2}}}{5} \theta + C \]
06
Back-Substitute the Trigonometric Expression
Since \( \tan(\theta) = \frac{u}{\sqrt{\frac{5}{2}}} \), then \( \theta = \tan^{-1}\left(\frac{u}{\sqrt{\frac{5}{2}}}\right) \). Substitute back to \( u = x+1 \):\[ \int \frac{1}{2x^2 + 4x + 7} dx = \frac{\sqrt{\frac{5}{2}}}{5} \tan^{-1}\left(\frac{x+1}{\sqrt{\frac{5}{2}}}\right) + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a technique used in algebra to simplify quadratic expressions, making them easier to work with during calculus operations like integration. The process involves transforming a quadratic polynomial of the form \(ax^2 + bx + c\) into a perfect square trinomial plus a constant. This helps in simplifying integrals, especially when dealing with rational functions.
Here's how you can complete the square:
Here's how you can complete the square:
- Factor out the leading coefficient if it is not 1. For instance, in the expression \(2x^2 + 4x + 7\), factor out 2 to get \(2(x^2 + 2x) + 7\).
- Next, complete the square for the expression inside the parentheses. To do this, take half of the coefficient of \(x\) (which is 2 in this example), square it, and add inside the bracket, then subtract it right outside. Therefore, \(x^2 + 2x\) becomes \((x+1)^2 - 1\).
- Substitute the newly formed perfect square into the expression: \(2((x+1)^2 - 1) + 7\). This simplifies to \(2(x+1)^2 + 5\).
Trigonometric Substitution
Trigonometric substitution is a powerful method used to evaluate integrals involving square roots and specific trigonometric identities. It involves substituting variables in an algebraic expression with trigonometric functions to transform the integral into a form that's easier to integrate. This method often works best when dealing with quadratic expressions under radicals or those resulting from completing the square.
Here's how it works in practice:
Here's how it works in practice:
- Choose a trigonometric substitution that matches the structure of the expression. For a term like \(2u^2 + 5\), where does not under a radical, consider \(u = \sqrt{\frac{5}{2}} \tan(\theta)\). This substitution simplifies the quadratic term using identities like \( \tan^2(\theta) + 1 = \sec^2(\theta)\).
- Substitute this into the integral, changing both the function and the differential. The differential \(du\) becomes \(\sqrt{\frac{5}{2}} \sec^2(\theta) d\theta\), and you end with an integral over \(\theta\).
- Simplify further using trigonometric identities to make the integration go smoothly, leading to easily integrable forms like \(\int d\theta\).
Definite and Indefinite Integrals
Integrals are a fundamental concept in calculus, representing the accumulation of quantities. There are two main types: definite integrals, which compute the area under a curve within a specified interval, and indefinite integrals, which represent a family of functions that differ by a constant.
Here's how they differ and overlap:
Here's how they differ and overlap:
- Indefinite integrals, denoted as \( \int f(x) dx \), seek the antiderivative of the function without specific limits of integration. The result includes a constant of integration, \(C\), as there are infinitely many antiderivatives.
- Definite integrals, denoted as \( \int_{a}^{b} f(x) dx \), calculate the net area between the function curve \(f(x)\) and the x-axis from \(x = a\) to \(x = b\). This form provides a numerical value, as opposed to a function.
- The Fundamental Theorem of Calculus links these two types, stating that if \(F(x)\) is the antiderivative of \(f(x)\), then \( \int_{a}^{b} f(x) dx = F(b) - F(a)\).
