Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^{2}}}=k\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right)+C\), where \(k=\) (A) 1 (B) 2 (C) 3 (D) 4

Step-by-Step Solution

Answer

The value of \( k \) is 2, so the answer is (B) 2.

Step 1: Substitution to simplify the integral

Substitute \( u = \sqrt{x} \). Then \( du = \frac{1}{2\sqrt{x}} dx \). So, \( x = u^2 \) and the denominator becomes \( 1+u \sqrt{u^2-u^4} = (1+u)\sqrt{u^2(1-u^2)} \).

Step 2: Simplify the integrand

The integrand becomes \( \int \frac{2u \, du}{(1+u)\sqrt{u^2(1-u^2)}} \), simplifying as \( \int \frac{2 du}{(1+u)\sqrt{1-u^2}} \).

Step 3: Use trigonometric substitution

Substitute \( u = \sin \theta \), then \( du = \cos \theta \, d\theta \) and the integrand becomes \( 2 \int \frac{d\theta}{1+\sin \theta} \).

Step 4: Solve the trigonometric integral

Use the identity \( 1 + \sin \theta = 2\cos^2(\theta/2) \), so the integral becomes \( 2 \int \frac{d\theta}{2\cos^2(\theta/2)} = \int \sec^2(\theta/2) d\theta \), which integrates to \( \tan(\theta/2) + C \).

Step 5: Convert back to original variable \( x \)

Since \( \tan(\theta/2) = \frac{\sin \theta}{1+\cos \theta} \) translates back to our substitution as \( \frac{u}{1 + \sqrt{1-u^2}} \), and since \( u = \sqrt{x} \), it further simplifies to \( \frac{\sqrt{x} - 1}{\sqrt{x} + 1} \).

Step 6: Determine the constant \( k \)

Comparing \( \tan(\theta/2) + C \) with \( k \left( \frac{\sqrt{x} - 1}{\sqrt{x} + 1} \right) + C \), we find that \( k = 2 \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus used to simplify integrals, making them easier to evaluate. It involves selecting a substitution that can transform the integral into a more manageable form. In this exercise, we start by substituting \( u = \sqrt{x} \).
This choice allows us to express the differential \( dx \) in terms of \( du \), specifically as \( dx = 2u \cdot du \). Consequently, the integral becomes more straightforward to handle.
The key advantage of substitution is that it reduces complexity by providing an alternative expression for the variables involved. Here, the original expression \((1 + \sqrt{x}) \sqrt{x-x^2}\) is transformed into \((1+ u)\sqrt{u^2(1-u^2)}\), which is much easier to manipulate.
By simplifying the problematic part of the integral, this approach paves the way for further simplification techniques or strategies, such as trigonometric substitution, enabling us to solve the integral efficiently.

Trigonometric Substitution

Trigonometric substitution is a clever technique used to evaluate integrals involving square roots, particularly those with expressions like \( \sqrt{1-x^2} \). This method leverages the identities of trigonometric functions to translate algebraic expressions into trigonometric ones, making them easier to integrate.
In the given problem, after applying the substitution \( u = \sqrt{x} \), we further simplify by setting \( u = \sin \theta \).
This choice is strategic, as \( \sqrt{1-u^2} \) becomes \( \cos \theta \) due to the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Once this transformation is complete, the integral:\[ 2\int \frac{d\theta}{1+\sin\theta} \] can be simplified using further trigonometric identities. One such identity used is \( 1 + \sin \theta = 2\cos^2(\theta/2) \), allowing the integral to further reduce to a form that is easier to solve. This application of trigonometric substitution provides a pathway to easily solve integrals that would otherwise be quite challenging.

Integration Techniques

Integration techniques are methods developed to tackle various forms of integrals, allowing us to grasp the seemingly complex task of integrating functions. The exercise above incorporates several integration techniques to arrive at the solution.
Firstly, substitution simplifies the complex integral into a more solvable form. Following this, trigonometric substitution further reduces the integral into a form delineated by easily manipulable trigonometric identities. Finally, the use of the identity \( 1 + \sin \theta = 2\cos^2(\theta/2) \) transforms the expression entirely into an integral involving \( \sec^2(\theta/2) \), which is straightforward to integrate.
This calculated maneuver enables us to evaluate the integral as \( \tan(\theta/2) + C \), which correlates to the original variable \( x \) via the series of substitutions made initially. These techniques demonstrated in this exercise exemplify the power of combining foundational calculus methods, affording a deterministic path to solving complex problems and determining constants like \( k \).

• The substitution method reduces complexity.
• Trigonometric substitution reshapes the problem.
• Utilizing trigonometric identities leads to final integration.

Mastering these techniques is key to solving intricate integration problems efficiently.