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Magazine Chapter 2: Problem 28
Evaluate \(\int_{-3}^{3} \frac{6}{9+x^{2}} d x\).
Short Answer
Expert verified
The value of the integral is \(\frac{\pi}{2}\).
Step by step solution
01
Identify the Integral
We need to evaluate the definite integral: \[\int_{-3}^{3} \frac{6}{9+x^{2}} \, dx.\]
02
Notice the Symmetric Limits
The integral is from \(-3\) to \(3\), which are symmetric around zero. For each \(x\) in \([-3, 0)\), there is a \(-x\) in \((0, 3]\). The function inside the integral, \(\frac{6}{9+x^2}\), is even since it depends on \(x^2\), which means \(f(x) = f(-x)\).
03
Apply Symmetry of Even Functions
For even functions over symmetric limits, the integral from \(-a\) to \(a\) simplifies by symmetry, doubling the integral from \(0\) to \(a\):\[\int_{-3}^{3} \frac{6}{9+x^{2}} \, dx = 2 \int_{0}^{3} \frac{6}{9+x^{2}} \, dx.\]
04
Find Antiderivative
The integral \(\int \frac{1}{a^2+x^{2}} \, dx\) is a standard form whose antiderivative is \(\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C\). In our problem, \(a^2 = 9\), so \(a = 3\). Multiply by 6:\[\int \frac{6}{9+x^{2}} \, dx = 2\tan^{-1}\left(\frac{x}{3}\right) + C.\]
05
Evaluate the Definite Integral
Using the antiderivative, evaluate the definite integral from 0 to 3:\[2 \left[ \tan^{-1}\left(\frac{x}{3}\right) \right]_{0}^{3} = 2 \left( \tan^{-1}(1) - \tan^{-1}(0) \right).\]
06
Calculate the Result
Evaluate the arctangent values:- \(\tan^{-1}(1) = \frac{\pi}{4}\) - \(\tan^{-1}(0) = 0\)\[2 \left( \frac{\pi}{4} - 0 \right) = \frac{\pi}{2}.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Even Function
A function is called even if it satisfies the condition that for every value of \(x\), \(f(x) = f(-x)\). This property implies that the graph of the function is symmetric with respect to the y-axis. In simpler terms, the left and right halves of the graph mirror each other.
- One common example of an even function is \(f(x) = x^2\), because \((x)^2 = (-x)^2\).
- Another example is \(f(x) = \cos x\) since \(\cos(-x) = \cos x\).
Symmetric Limits
When evaluating a definite integral, symmetric limits occur when the limits of integration are equidistant from zero but on opposite sides on the x-axis.
- For example, in \(\int_{-3}^{3} f(x) \, dx\), the limits \(-3\) and \(3\) are symmetric about zero.
- This symmetry can sometimes simplify the integration process, especially when dealing with even functions.
Antiderivative
Finding an antiderivative (or indefinite integral) is a process of determining a function whose derivative gives the original function you started with.
- For example, the antiderivative of \(f(x) = 2x\) is \(F(x) = x^2 + C\), where \(C\) is the constant of integration.
- Knowing common antiderivatives, such as those involving trigonometric functions, can be crucial in solving integrals.
Arctangent Function
The arctangent function is the inverse of the tangent function, denoted as \(\tan^{-1}(x)\) or \(\arctan(x)\). It measures an angle whose tangent is the given number.
- For instance, \(\tan^{-1}(1)\) results in \(\frac{\pi}{4}\).
- Similarly, \(\tan^{-1}(0)\) equals \(0\).
