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Magazine Chapter 7: Problem 51
Evaluate the integrals. \begin{equation}\int_{\ln (\pi / 6)}^{\ln (\pi / 2)} 2 e^{v} \cos e^{v} d v\end{equation}
Short Answer
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Step by step solution
01
Apply Substitution
To evaluate the integral \( \int_{\ln (\pi / 6)}^{\ln (\pi / 2)} 2 e^{v} \cos e^{v} \, dv \), we begin by making the substitution \( u = e^v \). Then, \( du = e^v dv \). Rearranging gives \( dv = \frac{du}{e^v} = \frac{du}{u} \).
02
Substitute and Adjust Limits
Once the substitution \( u = e^v \) is made, the integral becomes \( \int_{e^{\ln (\pi / 6)}}^{e^{\ln (\pi / 2)}} 2 \cos u \, du \) since \( e^{\ln(a)} = a \). Thus, the limits of integration change from \( \ln(\pi/6) \text{ to } \ln(\pi/2) \) to \( \pi/6 \text{ to } \pi/2 \).
03
Simplify the Integral
Now the transformed integral is \( \int_{\pi / 6}^{\pi / 2} 2 \cos u \, du \). This integral is straightforward to evaluate as it involves the cosine function.
04
Evaluate the Integral
The integral \( \int 2 \cos u \, du \) is equal to \( 2 \sin u + C \). Applying the limits of integration from \( \pi / 6 \) to \( \pi / 2 \), compute \[ 2 \sin(\pi / 2) - 2 \sin(\pi / 6) = 2 (1) - 2 (1/2) = 2 - 1 = 1. \].
05
Conclusion
The evaluated integral, after applying the limits, results in a final value of \( 1 \). Thus, the value of the original integral is \( 1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Definite integrals are a fundamental concept in integral calculus. They allow us to find the accumulated value, such as area under a curve between two points.
A definite integral is denoted as \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration, and \(f(x)\) is the function being integrated. Here are some important features of definite integrals:
A definite integral is denoted as \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration, and \(f(x)\) is the function being integrated. Here are some important features of definite integrals:
- Limits of Integration: \(a\) is the lower limit and \(b\) is the upper limit. These indicate the interval over which you are finding the area.
- Evaluation: The evaluation results in a numerical value, representing the net area between the function and the x-axis over the interval \([a, b]\).
- Properties: Definite integrals have properties like additivity and linearity, which help in simplifying complex integrations.
Integration by Substitution
Integration by substitution is a powerful technique used to simplify complex integrals. It works similarly to the chain rule for differentiation.
By introducing a substitution variable, we convert a difficult integral into a more manageable form. Here's how it works:
By introducing a substitution variable, we convert a difficult integral into a more manageable form. Here's how it works:
- Choose a Substitution: Substitute \(u = g(x)\) in the integral, where \(g(x)\) is part of the integrand making it complex.
- Differentiate: Find \(du = g'(x)dx\), then express \(dx\) in terms of \(du\) and \(x\) in terms of \(u\).
- Transform the Integral: Replace \(x\)-terms in the integral with \(u\)-terms, simplifying the integrand.
- Evaluate: Now solve this new integral in terms of \(u\), and later back-substitute the original variable if needed.
Cosine Function
The cosine function, denoted as \( \cos(x) \), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. Important aspects of the cosine function include:
- Periodicity: The cosine function is periodic with a period of \(2\pi\). This means it repeats its values every \(2\pi\) units.
- Range: It oscillates between -1 and 1, never exceeding these bounds. This trait influences the integrals involving \( \cos(x) \).
- Even Function: Cosine is an even function, meaning \( \cos(-x) = \cos(x) \). This symmetry is helpful in simplifying certain problems.
- Derivative and Integral: The derivative of \( \cos(x) \) is \( -\sin(x) \), and its integral is \( \sin(x) + C \). This last part helps in directly solving integrals involving the cosine function automatically.
Limits of Integration
Limits of integration define the interval over which we are calculating a definite integral. They are crucial because they specify the start and end points for the area calculation. Here's what is important to know:
- Upper and Lower Limits: Represented as the top and bottom numbers in the integral symbol, they determine where the integration starts and ends.
- Change with Substitution: When substitution is applied, these limits must be recalculated for the new variable, ensuring the intervals match the terms in the transformed integral.
- Mathematical Interpretation: The lower limit marks the initial point, and the upper limit marks the terminal point of integration along the x-axis, or in our substitution case, the u-axis.
- Evaluation Impact: Definite integrals are computed by evaluating the integral function at the upper limit and subtracting the evaluation at the lower limit \( F(b) - F(a) \).
