Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the integral \(\int_{0}^{\pi / 2} \frac{\sin 8 x \log (\cot x)}{\cos 2 x} d x\) is (A) \(\frac{1}{2}\) (B) \(\frac{-1}{2}\) (C) 1 (D) 0

Step-by-Step Solution

Answer

The value of the integral is 0.

Step 1: Simplify the Integral

Rewrite the integral expression. We have:\[I = \int_{0}^{\pi / 2} \frac{\sin 8x \log(\cot x)}{\cos 2x} \, dx\]Observe that the expression can benefit from a trigonometric identity for simplification, particularly focusing on using symmetry properties of trigonometric functions.

Step 2: Use Symmetry Properties

Consider the substitution technique using symmetry about \(x = \frac{\pi}{4}\). This is a common trick to handle definite integrals involving symmetric limits and periodic functions:Let \(u = \frac{\pi}{2} - x\), so that when \(x = 0, u = \frac{\pi}{2}\); and when \(x = \frac{\pi}{2}, u = 0\). The derivative \(dx = -du\), and rewrite the integral:\[I = \int_{\pi / 2}^{0} \frac{\sin(8(\frac{\pi}{2} - u)) \log(\cot(\frac{\pi}{2} - u))}{\cos(2(\frac{\pi}{2} - u))} \, (-du)\]Which becomes:\[I = \int_{0}^{\pi / 2} \frac{\sin(8u) \log(\tan u)}{-\cos(2u)} \, du = -\int_{0}^{\pi / 2} \frac{\sin(8u) \log(\tan u)}{\cos(2u)} \, du\]

Step 3: Compare Original Integral and Transformed Integral

Now, notice that:\[2I = \int_{0}^{\pi / 2} \frac{\sin(8x) \log(\cot x)}{\cos(2x)} \, dx + \int_{0}^{\pi / 2} \frac{\sin(8x) \log(\tan x)}{\cos(2x)} \, dx = \int_{0}^{\pi / 2} \frac{\sin(8x)(\log(\cot x) + \log(\tan x))}{\cos(2x)} \, dx = 0\]This occurs because \(\log(\cot x) + \log(\tan x) = \log(1) = 0\).Thus, simplify to find:\[2I = 0 \Rightarrow I = 0\]

Step 4: Conclude with the Integral Value

From the equations derived and the simplification using symmetry properties, the value calculated for the integral is:\[I = 0\]

Key Concepts

Integration Techniques

Integration is a fundamental technique in calculus that deals with the accumulation of quantities. When working with definite integrals, like \( \int_{a}^{b} f(x) \, dx \), we're interested in the area under the curve of a function \( f(x) \) between the bounds \( a \) and \( b \). Understanding and choosing the right integration technique is crucial to evaluating them efficiently.

In the given problem, simplifying the expression was a priority. Recognizing symmetry, substitutions, and trigonometric identities can streamline the process. In many scenarios, integrating by parts or partial fraction decomposition also aids in tackling complex integrals. Opt for integration by parts when your integrand is a product of two functions where one is easily differentiable and the other is easily integrable.

Partial fractions are often useful when dealing with rational functions. For trigonometric integrals specifically, identities can simplify expressions to functions that are more easily integrable by standard formulas.

Trigonometric Identities

Trigonometric identities form an essential toolkit for simplifying integrals that involve trigonometric functions. These identities express relationships between different trigonometric functions.

Some commonly used identities include:

• Pythagorean identities: \( \sin^2 x + \cos^2 x = 1 \)
• Angle sum and difference identities: \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)
• Double angle identities: \( \cos(2x) = \cos^2 x - \sin^2 x \)

In the problem at hand, trigonometric identities help in the evaluation by relating expressions within the integral. For instance, converting \( \log(\cot x) \) and \( \log(\tan x) \) can reveal underlying symmetries or cancellation that simplify the integral significantly. These identities are particularly helpful when combined with symmetry and substitution methods.

Substitution Method

The substitution method, also known as \( u \)-substitution, is a powerful technique used to simplify integrals. It involves replacing a part of the integrand with a single variable, \( u \), and transforming the differential \( dx \) accordingly.

To perform a substitution, follow these steps:

• Identify a substitution \( u = g(x) \) that simplifies the integrand to an easily integrable form.
• Compute \( du = g'(x) \, dx \).
• Substitute \( u \) and \( du \) into the integral and change the limits of integration if it's a definite integral.
• Integrate with respect to \( u \), then substitute back in terms of \( x \).

In this exercise, substituting \( u = \frac{\pi}{2} - x \) utilized the symmetry about the midpoint \( x = \frac{\pi}{4} \). This reflects around the center of the integral and sometimes can halve the work required or lead to powerful simplifications through direct cancellation.

Symmetry in Integrals

The concept of symmetry in integrals is a robust tool for simplifying their computation. Symmetry can often be used to directly evaluate integrals or significantly reduce the complexity of integration processes.

There are different types of symmetry to consider:

• Even and Odd Functions Symmetry: If \( f(x) \) is symmetric, the integral over symmetrical limits can be simplified. For example, \( \int_{-a}^{a} f(x) \, dx = 0 \) if \( f(x) \) is odd.
• Periodic Symmetry: Integrals over one period of a periodic function can often be determined by considering one cycle.

In the provided solution, symmetry was leveraged by substituting \( u = \frac{\pi}{2} - x \), reflecting the integral about \( x = \frac{\pi}{4} \). This transformation paired the original integral with a transformed version whose sum simplifies due to the combined symmetry \( \log(\cot x) + \log(\tan x) = 0 \). Recognizing such symmetry is a key step to converting a complicated integral into a more manageable form.