Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=\frac{x+2}{2 x+3}\), if \(\int\left(\frac{f(x)}{x^{2}}\right)^{1 / 2} d x\) \(=\frac{1}{\sqrt{2}} g\left(\frac{1+\sqrt{2 f(x)}}{1-\sqrt{2 f(x)}}\right)-\sqrt{\frac{2}{3}} h\left(\frac{\sqrt{3 f(x)}+\sqrt{2}}{\sqrt{3 f(x)-\sqrt{2}}}\right)+C\), then (A) \(g(x)=\log |x|\) (B) \(h(x)=\log |x|\) (C) \(g(x)=\tan ^{-1} x\) (D) \(h(x)=\tan ^{-1} x\)

Step-by-Step Solution

Answer

Correct options are (A) \( g(x)=\log |x| \) and (D) \( h(x)=\tan^{-1} x \).

Step 1: Understanding the Integral

The given problem involves an integral \( \int \left( \frac{f(x)}{x^{2}} \right)^{1/2} d x \) where \( f(x) = \frac{x+2}{2x+3} \).First, identify \( f(x) \) clearly. We know that the integral result involves functions \( g \) and \( h \), and we need to match them with given options.

Step 2: Simplifying \( \sqrt{f(x)} \)

Given \( f(x) = \frac{x+2}{2x+3} \), we calculate \( \sqrt{f(x)} = \sqrt{\frac{x+2}{2x+3}} \). To match terms in the integral form, it's crucial to understand this simplification as we'll use it in transformation functions.

Step 3: Analyzing the First Transformation Function

From the integral solution, identify \( 1 + \sqrt{2 f(x)} \).

Step 4: Analyzing the Second Transformation Function

Identify \( \frac{\sqrt{3 f(x)}+\sqrt{2}}{\sqrt{3 f(x)}-\sqrt{2}} \).

Step 5: Compare with Standard Results

By examining both transformations:- For \( g(x) \), convert expressions of type \( \frac{1+u}{1-u} \), which often result in logarithmic forms, i.e., \( g(x) = \log|x| \).- Similarly, expressions involving sums and differences of square roots and function inversions typically relate to inverse trigonometric functions.

Step 6: Conclusion

Observing transformations related to \( \log |x| \) and \( \tan^{-1} x \), identify that:- Given transformations suggest \( g(x) = \log |x| \) and \( h(x) = \tan^{-1} x \).Therefore, options (A) and (D) are correct.

Key Concepts

Integral Calculus

Integral Calculus is a major branch of calculus that deals with integrating functions to find accumulated quantities, such as area, volume, and length. It involves finding the antiderivative or integral of functions. Understanding definite integrals is crucial as they provide the accumulated value between two points.
In the given problem, we are tasked with evaluating the integral \( \int \left( \frac{f(x)}{x^2} \right)^{1/2} dx \). This expression represents the process of determining the accumulated area under the function \( \left( \frac{f(x)}{x^2} \right)^{1/2} \) over a specified interval.
The challenge lies in simplifying and solving these types of integrals because they may involve transformations that relate to known standard integrals used to find the results expressed in terms of functions like logarithms or inverse trigonometric functions.

Function Transformation

Function Transformation is an essential concept in calculus that involves changing or converting a function into another form to simplify integration or differentiation. Transformations can include shifting, stretching, reflecting, or rotating functions.
In solving this integral problem, several transformations are applied:

• Simplification: The original function \( f(x) = \frac{x+2}{2x+3} \) is simplified into a form usable for integration, \( \sqrt{f(x)} = \sqrt{\frac{x+2}{2x+3}} \).
• Expression Matching: The transformed expressions involving \( 1+\sqrt{2f(x)} \) and \( \frac{\sqrt{3f(x)}+\sqrt{2}}{\sqrt{3f(x)}-\sqrt{2}} \) are matched with standard forms to identify known function results.

These transformations help in recognizing and applying relevant formulas, such as logarithmic and trigonometric identities, to simplify and solve the integral.

Mathematical Functions

Mathematical Functions are the foundation of calculus and include a variety of standard and special functions that describe relationships between variables. Understanding these functions is critical for solving calculus problems involving differentiation and integration.
In this exercise, the functions \( g(x) \) and \( h(x) \) are expressed in terms of known mathematical functions, specifically:

• Logarithmic Functions: \( g(x) = \log|x| \) is often the result when expressions are transformed into \( \frac{1+u}{1-u} \), a form commonly encountered in integration.
• Inverse Trigonometric Functions: \( h(x) = \tan^{-1}x \) relates to expressions involving sums and differences of square roots that simplify into inverse trigonometric forms.

Identifying these transformations allows for the application of familiar mathematical functions to derive the solutions needed for the problem. This understanding is vital in connecting the integral with broader mathematical concepts.