Problem 100 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right) \text { is }}\) (A) \(x+\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c \mathrm{x}\) (B) \(x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (C) \(x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (D) \(x-\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c\)

Step-by-Step Solution

Answer

The value of the integral is option (D), \(x - \log|\cos(x-\frac{\pi}{4})| + C\).

Step 1: Transform the Integral

To solve \( \int \frac{\sin x \ d x}{\sin\left(x-\frac{\pi}{4}\right)} \), start by using the identity for sine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Thus, \( \sin\left(x - \frac{\pi}{4}\right) = \sin x \cos\frac{\pi}{4} - \cos x \sin\frac{\pi}{4} \). As \( \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} \), substitute this into the integral: \( \int \frac{\sin x}{\left(\sin x \cdot \frac{\sqrt{2}}{2} - \cos x \cdot\frac{\sqrt{2}}{2}\right)} dx \).

Step 2: Simplify the Expression

The expression can be rewritten using the common factor \( \frac{\sqrt{2}}{2} \): \( \int \frac{\sin x}{\frac{\sqrt{2}}{2}(\sin x - \cos x)} \, dx = \sqrt{2} \int \frac{\sin x}{\sin x - \cos x} \, dx \).

Step 3: Change of Variables

Let \( u = \sin x - \cos x \). Then the derivative is \( du = (\cos x + \sin x) dx \) or \( dx = \frac{du}{\cos x + \sin x} \). The integral becomes \( \sqrt{2} \int \frac{\sin x}{u} \cdot \frac{1}{\cos x + \sin x} \, du \). Notice that \( \cos x + \sin x = \sqrt{2} \sin\left(x+\frac{\pi}{4}\right) \), reducing the expression.

Step 4: Solve the Simplified Integral

The integral \( \int \frac{\sin x}{\sin(x-\pi/4)} \, dx \) simplifies to an integral of the form \( \int \frac{1}{u} \, du \). This integral evaluates to \( \log|u| + C \). Substitute back \( u = \sin x - \cos x \): \( \log|\sin x - \cos x| + C \).

Step 5: Final Expression Transformation

Recognizing \( \sin x - \cos x = \sqrt{2} \sin\left(x-\frac{\pi}{4}\right) \), the solution can be expressed as \( x - \log|\cos(x-\frac{\pi}{4})| + C \). This matches answer choice (D).

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique that is key when dealing with integrals involving trigonometric functions. These types of integrals often require clever manipulation of trigonometric identities to simplify the expression before integrating.
To tackle these integrals, one often utilizes fundamental trigonometric identities such as:

• 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: \( \sin(2x) = 2 \sin x \cos x \).

For example, in the given problem, we apply the sine difference identity to simplify \( \sin(x - \frac{\pi}{4}) \). This involves expressing \( \sin(x - \frac{\pi}{4}) \) in terms of \( \sin x \) and \( \cos x \) using the identity \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). By doing so, we can manipulate the integral into a more workable form, paving the way for further techniques like substitution.

Integration by Substitution

Integration by substitution is a method used to simplify the integration process by changing the variable of integration. This technique is akin to reversing the chain rule from differentiation. By substituting part of the integral with a new variable, we effectively transform the integral into a simpler form.
The general steps for integration by substitution are:

• Identify a portion of the integral to replace with a new variable \( u \) (e.g., \( u = \sin x - \cos x \)).
• Differentiating \( u \), find \( du \) and express it in terms of the original variable's differential \( dx \).
• Replace the identified portion and \( dx \) in the original integral with \( u \) and \( du \), respectively, thus obtaining a new integral in terms of \( u \).

In this exercise, after recognizing \( u = \sin x - \cos x \), we found that \( du = (\cos x + \sin x) dx \). Substituting these into the integral allows it to be expressed as \( \int \frac{1}{u} \, du \), which is simpler to compute. Once the integration is done, remember to substitute back the original variables to obtain the final answer in the terms of the original function variables.

Mathematical Identities

Mathematical identities are equations that hold true for all values within certain conditions. These identities provide foundational tools in calculus, especially in the integration of complex functions. Understanding and leveraging these identities can greatly simplify otherwise complicated expressions, making them easier to work with.
In trigonometric integration, several key identities are frequently used:

• Angle sum and difference identities: Useful for decomposing expressions into a form that fits standard integral templates.
• Pythagorean identities: Often help in converting trigonometric expressions into ones that are more easily integrable.
• Reduction formulas: Assist in breaking down complex higher-order trigonometric expressions.

Within the context of the presented problem, using the angle difference identity \( \sin(x - \frac{\pi}{4}) = \sin x \cdot \cos(\frac{\pi}{4}) - \cos x \cdot \sin(\frac{\pi}{4}) \) allows us to simplify and reframe the integral to a manageable form. These identities are an integral part of transforming and reducing expressions to facilitate the action of integration, underscoring their vital role in calculus.