Problem 136 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(I\) is the greatest of the definite integrals \(I_{1}=\int_{0}^{1} e^{-x} \cos ^{2} x d x, I_{2}=\int_{0}^{1} e^{-x^{2}} \cos ^{2} x d x\), \(I_{3}=\int_{0}^{1} e^{-x^{2}} d x\), and \(I_{4}=\int_{0}^{1} e^{-x^{2} / 2} d x\), then, (A) \(I=I_{1}\) (B) \(I=I_{2}\) (C) \(I=I_{3}\) (D) \(I=I_{4}\)

Step-by-Step Solution

Answer

(D) \(I = I_4\)

Step 1: Compare the Integrals

To determine which is the greatest among the integrals given, let us analyze their behavior over the interval \([0,1]\). All functions include damping factors due to exponential terms (\(e^{-x}\), \(e^{-x^2}\), and \(e^{-x^2/2}\)). The idea is to identify which integral encompasses the largest area under its curve based on these damping factors.

Step 2: Analyze Integral I1

The integral \(I_1 = \int_{0}^{1} e^{-x} \cos^2 x \, dx\) involves \(e^{-x}\), which decreases from 1 to \(e^{-1}\) and \(\cos^2 x\), which fluctuates between 0 and 1. Due to this fluctuation and the exponential decay, \(I_1\) is influenced by both factors.

Step 3: Analyze Integral I2

Similarly, \(I_2 = \int_{0}^{1} e^{-x^2} \cos^2 x \, dx\) involves \(e^{-x^2}\), which decreases less rapidly from 1 to \(e^{-1}\) at the end of the interval. The presence of \(\cos^2 x\) again dictates a fluctuating value, but the damping is less aggressive than with \(e^{-x}\).

Step 4: Analyze Integral I3

The integral \(I_3 = \int_{0}^{1} e^{-x^2} \, dx\) only involves the term \(e^{-x^2}\). Since there are no other multiplicative fluctuations (like the cosine term), this integral represents a more consistent value, potentially higher than \(I_2\) due to lack of fluctuations.

Step 5: Analyze Integral I4

Lastly, \(I_4 = \int_{0}^{1} e^{-x^2/2} \, dx\) features a damping factor \(e^{-x^2/2}\) which decreases even slower than \(e^{-x^2}\) over the interval. Thus, it is less restricted by exponential decay, potentially making this integral the greatest.

Step 6: Conclusion on Greatest Integral

From the above analysis, we see that \(I_3\) and \(I_4\) do not have fluctuations that \(I_1\) and \(I_2\) do. \(I_4\) has even less decay than \(I_3\), making \(I_4\) the greatest integral.

Key Concepts

Exponentials in Calculus

In calculus, exponential functions often feature prominently, especially when dealing with integrals. These functions typically have the form \(e^{-x}\), \(e^{-x^2}\), or \(e^{-x^2/2}\). The key characteristic of these exponential terms is their rapid decrease, or decay, over an interval. Understanding the behavior of these functions helps in predicting the properties of a given integral.

• For \(e^{-x}\): The function decreases rapidly from 1 to \(e^{-1}\) over the interval \([0, 1]\).
• For \(e^{-x^2}\): The decay is slower compared to \(e^{-x}\) as \(x^2\) increases slower than \(x\).
• For \(e^{-x^2/2}\): The decay is even less pronounced, as the factor \(-x^2/2\) causes a smaller rate of decrease.

This understanding is critical when comparing integrals with different exponential terms, as the size of the area under the curve can be significantly affected by how quickly or slowly these functions decrease.

Cosine Functions in Integration

Cosine functions, especially \(\cos^2(x)\), are common in integrals and bring unique properties due to their periodic nature. The function \(\cos^2(x)\) repeats every \(\pi\) and ranges between 0 and 1. This fluctuation plays an important role when multiplied by other functions in an integral, such as exponentials.

• In \(I_1\), \(\int_{0}^{1} e^{-x} \cos^2 x \, dx\), the rapid decay of the exponential \(e^{-x}\) is coupled with the fluctuating cosine term, creating a complex dynamic.
• In \(I_2\), \(\int_{0}^{1} e^{-x^2} \cos^2 x \, dx\), the slower decay \(e^{-x^2}\) is paired with the cosine fluctuation, leading to a slightly elevated integral value compared to \(I_1\).

The integration of cosine functions with damping factors like those found in exponentials illustrates how fluctuations impact the amount of area accumulated under a curve during integration.

Greatest Integral Evaluation

Finding the greatest integral involves evaluating each function's behavior over an interval, particularly focusing on the exponential damping and cosine fluctuation. In our original exercise, four integrals were compared to identify which provided the maximum area under its curve.

In analyzing integrals, it is crucial to consider the impact of both damping and fluctuation. For example, the integrals \(I_1\) and \(I_2\) include the damping factor of exponential decay and the added complexity of cosine fluctuations, reducing their overall area.

In contrast:

• \(I_3 = \int_{0}^{1} e^{-x^2} \, dx\) lacks cosine fluctuations, offering a steadier integration area.
• \(I_4 = \int_{0}^{1} e^{-x^2/2} \, dx\) not only avoids fluctuations but also features the least decay; it is the least restricted by exponential decay.

When determining the greatest integral, the absence of fluctuations and minimal decay in \(I_4\) suggested it encloses the largest area, making it the solution.