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

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 \( \frac{\pi}{2} \log 2 \) (Option D).

Step 1: Recognize the Integral Form

The integral \( \int_{0}^{1} \frac{8 \log (1+x)}{1+x^{2}} dx \) can involve the arctangent function, since \( \frac{1}{1+x^2} \) is the derivative of \( \arctan(x) \).

Step 2: Use Trigonometric Substitution

Consider substituting \( x = \tan(t) \), so \( dx = \sec^2(t) dt \). Then the integral bounds change as: when \( x = 0, t = 0 \) and when \( x = 1, t = \frac{\pi}{4} \).

Step 3: Substitute to Transform Integral

With \( x = \tan(t) \), the integral becomes \( \int_{0}^{\frac{\pi}{4}} 8 \log(1+\tan(t)) dt \) because \( 1+x^2 = \sec^2(t) \) and thus, the \( \sec^2(t) \) cancels with the denominator.

Step 4: Express \( \log(1+\tan(t)) \) in Simplified Form

Use trigonometric identities to simplify \( \log(1+\tan(t)) \) into \( \log(\sec(t) \cos(t) + \sin(t)) \). This can help rewrite the integral, but may require expansion using series or properties of logarithms.

Step 5: Evaluate the Integral

By utilizing known integrals or substitution, involving trigonometric identities, calculate the integral. For example, recognize forms that can be directly integrated to involve \( \log(\cos(t)) \) which often equals \( \sin^{-1}(t) \).

Step 6: Apply Limits and Solve

Evaluate at the bounds and simplify the expression. Simplify and substitute back terms, leading to a result involving \( \pi \).

Step 7: Check against provided options

After simplification, compare against provided answers, determining it matches \( \frac{\pi}{2} \log(2) \).

Key Concepts

Trigonometric Substitution

When dealing with integrals, especially those with polynomials in the denominator, trigonometric substitution can be a game-changer. This technique is useful for simplifying and transforming integrals into a more tractable form. Here, by substituting a trigonometric function for a variable, you can harness trigonometric identities to simplify the integration process.

For instance, consider substituting a variable, like by setting \( x = \tan(t) \). This is particularly effective in expressions involving \( 1 + x^2 \), because the identity \( 1 + \tan^2(t) = \sec^2(t) \) helps simplify the integrand.

• Replace \( x \) with \( \tan(t) \), turning expressions like \( 1+x^2 \) into simpler trigonometric identities.
• Also, change the differential \( dx \) into \( \sec^2(t) dt \), thus modifying the integral's domain accordingly.
• Overcome challenging integrations by using known trigonometric integrals and identities.

Learn to identify when a trigonometric substitution is applicable, and how it sheds light on the underlying function's behavior, which is invaluable in solving complex integrals.

Integration Techniques

Integration is one of the core techniques in calculus, often requiring creativity to solve. Various methods, such as substitution, integration by parts, and partial fractions, equip you to tackle diverse problems effectively.

For integrals involving logarithmic or trigonometric components, turning to specific methods can simplify your task.

• Substitution: Useful when the integral includes composite functions, allowing you to simplify inner functions.
• Partial Fraction Decomposition: Breaks down rational functions for simpler integration.
• Recognition: Identify integral forms like \( \frac{1}{1+x^2} \), which ties directly to the derivative of \( \arctan(x) \).

When attempting exercise solutions, choose the technique aligning best with the integral's structure.
Mastering these strategies not only helps solve textbook problems but builds a foundation for tackling real-world mathematical challenges.

Arctangent Function

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function. It provides the angle whose tangent is \( x \), sitting prominently in calculus due to its simple derivative and integral properties.

The integral \( \int \frac{1}{1+x^2} dx \) directly results in \( \arctan(x) + C \), highlighting its importance in integration exercises.

• Simplification: Recognize parts of integrands that resemble arctangent and utilize this to simplify integrals.
• Derivatives: Since the derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \), spotting this within integrals can help solve them efficiently.
• Application: Common in physics and engineering for resolving angles and approximating functions.

The clarity and utility of the arctangent function make it essential for students to understand when manipulating trigonometric integrals.