Problem 198 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(l_{1}=\int_{0}^{1} 2^{x^{2}} d x, l_{2}=\int_{0}^{1} 2^{x^{2}} d x, l_{3}=\int_{1}^{2} 2^{x^{2}} d x\), and \(l_{4}=\int_{1}^{2} 2^{x^{1}} d x\) then (A) \(l_{2}>l_{1}\) (B) \(l_{1}>l_{2}\) (C) \(l_{3}=l_{4}\) (D) \(l_{3}>l_{4}\)

Step-by-Step Solution

Answer

The correct choice is (D) \( l_{3} > l_{4} \).

Step 1: Identify the Integrals

First, identify the two types of integrals given in the problem statement. We have \( l_{1} = \int_{0}^{1} 2^{x^2} \, dx \) and \( l_{2} = \int_{0}^{1} 2^{x^2} \, dx \), \( l_{3} = \int_{1}^{2} 2^{x^2} \, dx \), and \( l_{4} = \int_{1}^{2} 2^{x} \, dx \). Recognize that \( l_1 \) and \( l_2 \) are the same as \( \int_{0}^{1} 2^{x^2} \, dx \) and need no comparison since they are equal.

Step 2: Compare l_3 and l_4

Now consider \( l_3 = \int_{1}^{2} 2^{x^2} \, dx \) and \( l_4 = \int_{1}^{2} 2^{x} \, dx \). We are interested to see if \( l_3 = l_4 \) or \( l_3 > l_4 \). Here, the expressions inside the integrals differ. For \( l_3 \), we have an exponential of \( x^2 \), which grows faster than the exponential of \( x \). Therefore, \( 2^{x^2} > 2^x \) for \( x > 1 \). This suggests that the integral from 1 to 2 of \( 2^{x^2} \) is greater than the integral of \( 2^x \).

Step 3: Conclusion

From Step 1, since \( l_1 = l_2 \), options A and B are incorrect. From Step 2, the result indicates \( l_3 > l_4 \), and therefore, option C is incorrect. Hence, the correct choice is option D.

Key Concepts

Definite Integrals

Definite integrals represent the accumulation of quantities and provide the total value of a function across an interval. In mathematical terms, a definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \). This represents the area under the curve of the function \( f(x) \) from the point \( a \) to \( b \). Understanding the boundaries, or limits, \( a \) and \( b \), is crucial as they define the range over which you are integrating.
For example, in the problem given, \( l_1 = \int_{0}^{1} 2^{x^2} \, dx \) and \( l_3 = \int_{1}^{2} 2^{x^2} \, dx \), the integration occurs over two different intervals: from 0 to 1 and from 1 to 2, respectively. These intervals affect the outcome of the integrals significantly. The definite integral also has practical applications in fields like physics and engineering, as it allows for the computation of quantities such as distance, area, and work based on varying functions.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They grow rapidly and are of the form \( f(x) = a^{x} \), where \( a \) is a positive constant. This can either involve an exponent that is a simple variable, like \( 2^x \), or a more complex expression, like \( 2^{x^2} \).
Exponential functions are significant in this exercise because they define the rate of growth of the integrands for the integrals \( l_3 \) and \( l_4 \). It's important to note that an exponential function with a quadratic exponent, such as \( 2^{x^2} \), increases much faster than one with a linear exponent, such as \( 2^x \).
Key properties to remember about exponential functions are:

• They are always positive for all real number inputs.
• Their rate of growth is directly affected by the exponent.
• In comparing two exponential functions, the function with the higher exponent often grows significantly faster.

In our problem, this explains why the integral \( l_3 = \int_{1}^{2} 2^{x^2} \, dx \) results in a larger value than \( l_4 = \int_{1}^{2} 2^{x} \, dx \).

Comparison of Functions

When comparing functions to determine which grows faster or produces a larger integral output, it's important to look closely at the behavior of their exponents and coefficients.
In this specific exercise, comparing \( 2^{x^2} \) and \( 2^x \) involves understanding how these functions behave over the interval \( x \geq 1 \). Since \( x^2 \) grows larger than \( x \) as \( x \) increases, it leads to \( 2^{x^2} \) always being greater than \( 2^x \) within the interval from 1 to 2. This is a key observation that helps in determining the overall magnitude of the integrals \( l_3 \) and \( l_4 \).
To effectively compare functions in other similar contexts:

• Look at the degree and the type of exponents in the functions.
• Understand their behavior over the given interval.
• Consider the specific application of each function in practical problems.

These steps can provide a better visual and analytical understanding, enabling accurate judgments about which functions yield larger definite integrals.