Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\int_{0}^{1} \frac{8 \log (1+x)}{1+x^{2}} d x\) is (A) \(\log 2\) (B) \(\pi \log 2\) (C) \(\frac{\pi}{8} \log 2\) (D) \(\frac{\pi}{2} \log 2\)

Step-by-Step Solution

Answer

The value of the integral is \(\pi \log 2\), which corresponds to option (B).

Step 1: Identify the Integral

The given integral is \( \int_{0}^{1} \frac{8 \log (1+x)}{1+x^{2}} \, dx \). Our objective is to evaluate this expression and identify the correct answer from the provided options.

Step 2: Use Substitution

To solve the integral, we can use the substitution \( x = \tan{\theta} \). This transforms the limits: when \( x = 0 \), \( \theta = 0 \); and when \( x = 1 \), \( \theta = \frac{\pi}{4} \). The derivative \( dx = \sec^2{\theta} \, d\theta \). Thus, the integral becomes:\[ \int_{0}^{\frac{\pi}{4}} \frac{8 \log{(1 + \tan{\theta})}}{1 + \tan^2{\theta}} \sec^2{\theta} \, d\theta. \]

Step 3: Simplify Using Trigonometric Identities

Using the identity \( 1 + \tan^2{\theta} = \sec^2{\theta} \), the integral simplifies to:\[ \int_{0}^{\frac{\pi}{4}} 8 \log{(1 + \tan{\theta})} \, d\theta. \]

Step 4: Further Simplification

Note that \( 1 + \tan{\theta} = \frac{1 + \sin{\theta}}{\cos{\theta}} = \sec{\theta} + \tan{\theta} \). Therefore, the integral simplifies to:\[ \int_{0}^{\frac{\pi}{4}} 8 (\log{\sec{\theta}} + \log{\tan{\theta}}) \, d\theta. \]

Step 5: Separate and Solve the Integral

Split the integral into two parts:\[ 8 \int_{0}^{\frac{\pi}{4}} \log{\sec{\theta}} \, d\theta + 8 \int_{0}^{\frac{\pi}{4}} \log{\tan{\theta}} \, d\theta. \]

Step 6: Evaluate Each Integral

The integral \( \int_{0}^{\frac{\pi}{4}} \log{\sec{\theta}} \, d\theta \) and \( \int_{0}^{\frac{\pi}{4}} \log{\tan{\theta}} \, d\theta \) are well-known and both equal to \( \frac{\pi}{8} \log{2} \).

Step 7: Add the Results

Add the results from the two integrals:\[ 8 \left( \frac{\pi}{8} \log{2} + \frac{\pi}{8} \log{2} \right) = \pi \log{2}. \]

Step 8: Compare with Options

Compare the final result \( \pi \log{2} \) with the options. It matches option (B) \( \pi \log 2 \).

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus for solving complex integration problems. In the given exercise, we need to evaluate a definite integral \( \int_{0}^{1} \frac{8 \log (1+x)}{1+x^{2}} \, dx \).
When dealing with integrals, especially those that appear complicated, it's helpful to consider different techniques, such as:

• Substitution
• Integration by parts
• Partial fraction decomposition

Each of these can simplify the problem, but choosing the right one is key. In our problem, the trick lies in picking trigonometric substitution, as it transforms the integral into a more manageable form. Understanding which technique to apply comes with practice and familiarity with various integral forms.

Trigonometric Substitution

Trigonometric substitution is a clever method to simplify integrals involving square roots or quadratic expressions. By using trigonometric identities, we can convert the integral into a form that is easier to evaluate.
In this exercise, we substitute \( x = \tan{\theta} \), which changes the limits accordingly:

• When \( x = 0 \), \( \theta = 0 \).
• When \( x = 1 \), \( \theta = \frac{\pi}{4} \).

Additionally, the differential changes as \( dx = \sec^2{\theta} \, d\theta \).
This transformation is pivotal because it simplifies the expression \( 1 + x^2 = \sec^2{\theta} \), allowing the integral \( \int 8 \log{(1+\tan{\theta})} \, d\theta \) to be further broken down and solved using integration techniques resulting in well-known integrals.

Integral Calculus

Integral Calculus is the branch of mathematics focusing on integrals and their applications.
When evaluating definite integrals, the process involves finding an antiderivative and applying the limits to determine the area under a curve. In this specific exercise, once we simplified the integral using trigonometric substitution, we needed to solve:

• \( 8 \int_{0}^{\frac{\pi}{4}} \log{\sec{\theta}} \, d\theta \)
• \( 8 \int_{0}^{\frac{\pi}{4}} \log{\tan{\theta}} \, d\theta \)

Both these integrals have known solutions, calculated as \( \frac{\pi}{8} \log{2} \).
Adding the results gives the final answer of \( \pi \log{2} \), showing how integral calculus, combined with techniques like trigonometric substitution, allows for solving complex problems systematically. Integral calculus enables you to work with various kinds of functions to find different areas and accumulated quantities.