Problem 105
Question
\(\int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}}\) lies between (A) \(\frac{\pi}{6 \sqrt{3}}\) and \(\frac{\pi}{4}\) (B) \(\frac{\pi}{3 \sqrt{3}}\) and \(\frac{\pi}{2}\) (C) \(\frac{\pi}{3 \sqrt{3}}\) and \(\frac{\pi}{4}\) (D) None of these
Step-by-Step Solution
Verified Answer
The correct option is (B).
1Step 1: Simplify the Expression
First, examine the integrand \( \frac{1}{1+x^2+2x^5} \). Notice that it's not a standard form and it's complicated for direct integration. Therefore, consider whether it's possible to simplify the expression to estimate the bounds of the integral.
2Step 2: Estimate the Bounds
Determine bounds on \( \frac{1}{1+x^2+2x^5} \). Since \( x^5 \) becomes very small for \( x \leq 1 \), we can approximate the function between simple forms: \( \frac{1}{1+x^2} \leq \frac{1}{1+x^2+2x^5} \leq \frac{1}{1} = 1 \).
3Step 3: Integrate Simple Bounds
Integrate \( \frac{1}{1+x^2} \) and \( 1 \) from 0 to 1 to set the range for the integral:\[ \int_{0}^{1} \frac{1}{1+x^2} \, dx = \left[ \tan^{-1}(x) \right]_{0}^{1} = \frac{\pi}{4} \]and \[ \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 \].
4Step 4: Compare with Given Options
We have calculated that our integral is between \( \frac{\pi}{4} \, \text{(lower bound)} \) and \( 1 \, \text{(upper bound)} \). Compare this to the given options:- Option (A): \( \frac{\pi}{6 \sqrt{3}} \approx 0.303 \) and \( \frac{\pi}{4} \approx 0.785 \)- Option (B): \( \frac{\pi}{3 \sqrt{3}} \approx 0.605 \) and \( \frac{\pi}{2} \approx 1.57 \)- Option (C): \( \frac{\pi}{3 \sqrt{3}} \approx 0.605 \) and \( \frac{\pi}{4} \approx 0.785 \)- Here, only Option (B) encompasses the range \( \left( \frac{\pi}{4}, 1 \right) \).
5Step 5: Conclude the Correct Option
Based on our calculations, the integral \( \int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}} \) lies between \( \frac{\pi}{3 \sqrt{3}} \) and \( \frac{\pi}{2} \). Thus, the correct option is (B).
Key Concepts
Definite IntegralsApproximation MethodsIntegration Bounds
Definite Integrals
Definite integrals are integral calculations with set limits, from which one can determine the exact value of the function within specific bounds. Here, we are dealing with the integral \[\int_{0}^{1} \frac{d x}{1+x^{2}+2x^{5}}\]which involves finding the area under the curve of the function from 0 to 1.
Understanding definite integrals includes comprehending that they are about the summation of infinitesimal quantities over a range. They play a crucial role in several areas of calculus and often relate to physical quantities. The limits of integration define the interval over which the function is integrated, providing a specific numerical result instead of an expression with constants.
When you solve a definite integral, you can visualize it as the part of the plane that the curve of the function encloses between the limits you set.
Understanding definite integrals includes comprehending that they are about the summation of infinitesimal quantities over a range. They play a crucial role in several areas of calculus and often relate to physical quantities. The limits of integration define the interval over which the function is integrated, providing a specific numerical result instead of an expression with constants.
When you solve a definite integral, you can visualize it as the part of the plane that the curve of the function encloses between the limits you set.
Approximation Methods
Sometimes, it is challenging to integrate a complex function directly. This is where approximation methods step in. For our problem, the integrand\[\frac{1}{1+x^2+2x^5}\]is complicated to solve precisely, so we leverage simpler bounds to approximate its value.
Approximation methods help to set boundaries for more complex functions that may not have straightforward solutions:
Approximation methods help to set boundaries for more complex functions that may not have straightforward solutions:
- Identify simpler functions that the complex function can be compared to; here, we're using \(\frac{1}{1+x^2}\) and 1.
- Use these approximate forms to find simple integrals that can act as bounds for your more difficult integral.
Integration Bounds
Integration bounds are the specified points that determine the section of the curve to be evaluated. In our exercise, these are 0 and 1, indicating where our function should be considered.
These limits are essential because they define:
Bounds help isolate the behavior of the function within a range; see how \(x^5\) becomes less significant compared to 1 or \(x^2\) as \(x\) ranges from 0 to 1. This insight simplifies our task of setting accurate approximations and comprehending how these bounds influence the function.
These limits are essential because they define:
- The starting and ending points of the area we’re interested in calculating.
- What portion of the curve's total area is captured by the definite integral.
Bounds help isolate the behavior of the function within a range; see how \(x^5\) becomes less significant compared to 1 or \(x^2\) as \(x\) ranges from 0 to 1. This insight simplifies our task of setting accurate approximations and comprehending how these bounds influence the function.
Other exercises in this chapter
Problem 103
If \(I_{1}=\int_{0}^{a}[x] d x\) and \(I_{2}=\int_{0}^{a}\\{x\\} d x\), where \([x]\) and \(\\{x\\}\) denote, respectively, the integral and fractional parts of
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