Problem 57 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\left.\right|_{10} ^{19} \frac{\sin x d x}{1+x^{8}} \mid\) is less than (A) \(10^{-10}\) (B) \(10^{-11}\) (C) \(10^{-7}\) (D) \(10^{-9}\)

Step-by-Step Solution

Answer

(C) \(10^{-7}\).

Step 1: Understand the Problem

The expression given is \(\left| \int_{10}^{19} \frac{\sin x}{1+x^8} \, dx \right|\). We need to find an upper bound for this integral to determine which option it is less than.

Step 2: Estimating \(\sin x\)

The function \(\sin x\) oscillates between -1 and 1 for all \(x\). Thus, we have \(|\sin x| \leq 1\).

Step 3: Simplify the Integrand

Since \(|\sin x| \leq 1\), we can bound the integrand by \(\left| \frac{\sin x}{1+x^8} \right| \leq \frac{1}{1+x^8}\).

Step 4: Estimating the Integral

Now, we estimate \(\int_{10}^{19} \frac{1}{1+x^8} \, dx\). Since \(1+x^8\) is increasing over \([10,19]\), we estimate it further using its value at \(x=10\) and \(x=19\).

Step 5: Calculate Upper Bound of Integral

At \(x=10\), \(1+x^8 = 1+10^8\) and at \(x=19\), \(1+x^8 = 1+19^8\). Thus:\[\int_{10}^{19} \frac{1}{1+x^8} \, dx \leq \int_{10}^{19} \frac{1}{10^8} \, dx = \frac{1}{10^8} \times (19 - 10) = \frac{9}{10^8} = 9 \times 10^{-8}\]

Step 6: Compare and Choose the Correct Option

The calculated upper bound \(9 \times 10^{-8}\) is still greater than \(10^{-9}\) and \(10^{-10}\), but less than \(10^{-7}\). Thus, the correct answer is option (C), \(10^{-7}\).

Key Concepts

Integral Bounds

When working with definite integrals, finding the integral bounds is crucial for understanding the region under consideration. Integral bounds define the lower and upper limits of the integration, often denoted as \([a, b]\). In this problem, the integral bounds are from 10 to 19. These bounds tell us that we are examining the behavior of the function \(\frac{\sin x}{1+x^8}\) within this interval.

To work with integral bounds effectively:

• Clearly identify the interval: Ensure you are integrating over the correct range by checking the limits carefully.
• Understand the implications: The bounds determine the segment of the function you will be analyzing. Sometimes, certain properties of a function, like periodicity in trigonometric functions, need consideration within these bounds.
• Estimation: Often, when exact calculation is complex, you can estimate the integral using properties of the function within those bounds (as we do by bounding the integrand).

Thus, in this scenario, the bounds help us encapsulate the function's behavior between a specific range.

Trigonometric Functions

Trigonometric functions like \(\sin x\) play a significant role in many calculus problems, including integrals. Understanding their properties helps in simplifying and estimating integrals.

For \(\sin x\):

• Periodicity: \(\sin x\) oscillates with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units.
• Range: The function has a range of [-1, 1], which means \(\sin x\) does not go beyond these values regardless of \(x\).
• Symmetry and behavior: It is symmetrical about the origin and varies smoothly between its extreme values.

In this estimation problem, because \(\sin x\) varies between -1 and 1, it allows us to bound the integrand by noting \(|\sin x| \leq 1\). This provides a simpler expression for approximation since \(|\sin x|\)'s effect can be simplified.

Inequality Estimation

Inequality estimation is a powerful tool in calculus, allowing us to approximate the values and bounds of integrals. By using inequalities, we can approximate the behavior of complex functions in a simpler form.

Here's how it works for this exercise:

• Bounding the integrand: We take \( \left| \frac{\sin x}{1+x^8} \right| \) and note it can be bounded by \( \frac{1}{1+x^8} \) since \(|\sin x| \leq 1\).
• Estimating the integral: Knowing that \( \frac{1}{1+x^8} \) is a decreasing function over the interval [10, 19], we integrate it to approximate the behavior of the original integral. This uses the property that the larger \(x\) becomes, the smaller the expression \(1+x^8\) becomes, tightening the bound of the integral.
• Comparison with options: After estimation, this bound can be compared to the multiple-choice options to find the closest match. This involves simple arithmetic and reasoning about the computed bound using inequality estimates.

Utilizing inequality estimation helps not only in solving the problem but also in understanding the integral's behavior over given bounds, leading to faster and computationally easier solutions.