Problem 104 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}}\) lies between (A) \(\frac{1}{4}\) and (B) \(\frac{1}{4}\) and \(\frac{1}{2}\) (C) \(\frac{1}{2}\) and 1 (D) None of these

Step-by-Step Solution

Answer

The integral result lies between \(\frac{1}{4}\) and \(\frac{1}{2}\), so the answer is option (B).

Step 1: Simplify the Integrand Expression

The given integral is \(\int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}}\). We start by simplifying the expression in the denominator: \(1 + x^2 + 2x^5\). As \(x\) ranges from 0 to 1, both \(x^2\) and \(2x^5\) will also vary from 0 to their respective maxima of 1 and 2. Therefore, the denominator varies from 1 (when \(x=0\)) to a maximum of approximately 4 (when \(x=1\)).

Step 2: Evaluate Estimations for the Integrand Function

To estimate the integral, we consider the bounds by observing the function behavior. The integrated function \(\frac{1}{1 + x^2 + 2x^5}\) is between \(\frac{1}{1+0+0} = 1\) and \(\frac{1}{1+1+2} = \frac{1}{4}\), depending on \(x\). Since \(\frac{1}{1+x^2+2x^5}\) decreases from 1 to approximately \(\frac{1}{4}\), the average behavior over \([0,1]\) will give a median estimate between \(\frac{1}{4}\) and a little less than \(\frac{1}{2}\).

Step 3: Choose the Range for the Integral's Result

Now, we assess the available multiple-choice options. The region where the calculated integral can reasonably fall, given our findings, is between \(\frac{1}{4}\) and \(\frac{1}{2}\). The choice that covers this estimation is option (B): It is between \(\frac{1}{4}\) and \(\frac{1}{2}\).

Key Concepts

Definite Integral

A definite integral is a fundamental concept in integral calculus, used to calculate the area under a curve over a specific interval. In mathematical terms, the definite integral of a function \( f(x) \) from \( a \) to \( b \) is represented as \[ \int_{a}^{b} f(x) \; dx. \] This process involves two main parts:

• Finding the antiderivative (or integral) of \( f(x) \).
• Calculating the net area by subtracting the value of the antiderivative at the lower limit \( a \) from its value at the upper limit \( b \).

In the given problem, the definite integral is set up to find the result of integrating the function \( \frac{1}{1+x^{2}+2x^{5}} \) from 0 to 1. This determines the accumulated area between the function and the x-axis from \( x=0 \) to \( x=1 \). Understanding definite integrals is key to solving complex calculus problems involving real-world scenarios such as calculating distances, areas, and even the accumulation of quantities.

Integrand Simplification

In calculus, simplifying the integrand, which is the function being integrated, is often necessary for easier evaluation of an integral. A simpler form of the integrand makes it easier to estimate, analyze, and solve. In the problem at hand, the integrand is initially given as \( \frac{1}{1+x^{2}+2x^{5}} \). The simplification process involves analyzing the components of the denominator, \( 1 + x^2 + 2x^5 \), over the interval \([0,1]\).

• At \( x = 0 \), both \( x^2 \) and \( 2x^5 \) are 0, making the denominator equal to 1.


• At \( x = 1 \), \( x^2 \) becomes 1 and \( 2x^5 \) becomes 2, for a maximum total of 4 in the denominator.

This evaluation shows how the integrand varies; it starts from \( 1 \) and reduces to \( \frac{1}{4} \) as \( x \) approaches 1. Identifying such ranges facilitates the estimation of the integral by narrowing down the complexity of the initial expression.

Function Estimation

Function estimation is crucial when calculating integrals, especially when finding the exact integral is either complex or not possible through elementary means. By estimating, you can reasonably guess the integral's value based on the behavior of the function over the interval.For the given problem, the function \( \frac{1}{1+x^{2}+2x^{5}} \) behaves differently at different points of \([0, 1]\):

• The function value is highest at \( x = 0 \), where it equals 1.


• It decreases as \( x \) approaches 1, reaching approximately \( \frac{1}{4} \).

When estimating, these evaluations point to the integral's average value residing between the bounds of \( \frac{1}{4} \) and slightly less than \( \frac{1}{2} \). Recognizing this range helps in choosing the correct answer, as the integral will fall within this estimate – making option (B) the most reasonable choice. Such estimation techniques are vital in calculus for approximating solutions where precision calculations are challenging.