Problem 37
Question
Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 2}^{\pi / 2} \frac{\sin x}{1+\cos x} d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to 0 due to the symmetry of the odd function.
1Step 1: Analyze Symmetry of the Integral
The function \( f(x) = \frac{\sin x}{1+\cos x} \) is odd because \( \sin(-x) = -\sin x \) and \( \cos(-x) = \cos x \). Thus, \( f(-x) = -f(x) \).
2Step 2: Use Symmetry Property of Odd Functions
For any odd function \( f(x) \), the integral over a symmetric interval around zero (\( -a \) to \( a \)) is zero: \[ \int_{-a}^{a} f(x) \, dx = 0 \].
3Step 3: Apply the Symmetry Property to the Given Integral
Since the function \( \frac{\sin x}{1+\cos x} \) is odd, the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sin x}{1+\cos x} \, dx = 0 \).
Key Concepts
Odd FunctionsSymmetric IntervalProperties of Integrals
Odd Functions
Odd functions are a special type of function with a unique symmetry property. A function is called odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in the function's domain. This essentially means if you take the negative of the input, the output will also be the negative of the initial output. Recognizing odd functions is crucial, especially when dealing with integrals over symmetric intervals.
For example, \( \sin x \) is an odd function since \( \sin(-x) = -\sin x \). Understanding this property helps in solving integrals more quickly, especially in scenarios where the symmetry can be utilized to simplify calculations.
For example, \( \sin x \) is an odd function since \( \sin(-x) = -\sin x \). Understanding this property helps in solving integrals more quickly, especially in scenarios where the symmetry can be utilized to simplify calculations.
Symmetric Interval
A symmetric interval refers to an interval that is evenly distributed around zero, such as \( [-a, a] \). When considering integrals, these symmetric intervals can offer significant shortcuts, especially when combined with functions that exhibit specific symmetry properties, like odd or even functions.
The importance of symmetric intervals becomes more evident when applied with odd functions. For an odd function integrated over a symmetric interval, the result is zero. This is because the areas above and below the x-axis cancel each other out perfectly. When you look at the integral \[ \int_{-a}^{a} f(x) \, dx \],if \( f(x) \) is odd, it simplifies your calculation entirely as the result is 0.
In our exercise, the interval \( [-\pi/2, \pi/2] \) is symmetric around zero, allowing us to apply this property effectively.
The importance of symmetric intervals becomes more evident when applied with odd functions. For an odd function integrated over a symmetric interval, the result is zero. This is because the areas above and below the x-axis cancel each other out perfectly. When you look at the integral \[ \int_{-a}^{a} f(x) \, dx \],if \( f(x) \) is odd, it simplifies your calculation entirely as the result is 0.
In our exercise, the interval \( [-\pi/2, \pi/2] \) is symmetric around zero, allowing us to apply this property effectively.
Properties of Integrals
Integrals have several important properties that can be quite helpful in simplifying complex problems. Some of these properties are crucial in evaluating integrals using symmetry, especially with odd and even functions.
- Linearity: The integral of a sum is equal to the sum of the integrals.
- Interval Splitting: An integral over an interval can be split into multiple integrals over sub-intervals.
- Symmetry: For odd functions, the integral over a symmetric interval is zero. This property is very powerful because it allows us to solve some integrals instantly.
Other exercises in this chapter
Problem 36
Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a s
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Let \(x_{1}, x_{2}, \ldots, x_{n}\) be any real numbers. Find the value of \(c\) that minimizes \(\sum_{i=1}^{n}\left(x_{i}-c\right)^{2}\).
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use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{3} \frac{1}{(t+2)^{2}} d t $$
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Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a s
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