Problem 226 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of the integral \(\int_{0}^{\pi} \sqrt{1+4 \sin ^{2} \frac{x}{2}-4 \sin \frac{x}{2}} d x\) equals [2014] (A) \(\pi-4\) (B) \(\frac{2 \pi}{3}-4-4 \sqrt{3}\) (C) \(4 \sqrt{3}-4\) (D) \(4 \sqrt{3}-4-\frac{\pi}{3}\)

Step-by-Step Solution

Answer

The integral value is \(\pi - 4 + 2\sqrt{3}\). None of the provided options match exactly.

Step 1: Substitution Preparation

First, we need to make the expression under the square root more manageable. Recognize that \( \sin^2 \frac{x}{2} \) can be expressed as \( 1 - \cos^2 \frac{x}{2} \). This step simplifies calculations by changing the trigonometric term.

Step 2: Simplify the Expression

Recognize that the integrand is of the form \[ \int_{0}^{\pi} \sqrt{(1 - 2\sin \frac{x}{2})^2} \, dx \]Since the square root of a square is the absolute value, replace the expression with its absolute value:\[ \int_{0}^{\pi} |1 - 2\sin \frac{x}{2}| \, dx \]

Step 3: Solve the Absolute Value

The expression \(1 - 2 \sin \frac{x}{2} \) equals zero if \( \sin \frac{x}{2} = \frac{1}{2} \), i.e., at \( \frac{x}{2} = \frac{\pi}{6} \) or \( x = \frac{\pi}{3} \). Thus, split the integral at \( \frac{\pi}{3} \):\[ \int_{0}^{\frac{\pi}{3}} (2\sin \frac{x}{2} - 1) \, dx + \int_{\frac{\pi}{3}}^{\pi} (1 - 2\sin \frac{x}{2}) \, dx \]

Step 4: Calculate Each Integral

Evaluate each integral separately:1. For \( \int_{0}^{\frac{\pi}{3}} (2\sin \frac{x}{2} - 1) \, dx \): Use substitution \( u = \sin \frac{x}{2} \) to solve.2. For \( \int_{\frac{\pi}{3}}^{\pi} (1 - 2\sin \frac{x}{2}) \, dx \): Similarly use substitution \( u = \sin \frac{x}{2} \) to solve. Compute the definite integrals.

Step 5: Integrate and Combine Results

Carry out the integration:\( \int_{0}^{\frac{\pi}{3}} 2\sin \frac{x}{2} - 1 \, dx = \frac{\pi}{3} - 1 \) and\( \int_{\frac{\pi}{3}}^{\pi} 1 - 2\sin \frac{x}{2} \, dx = \frac{2\pi}{3} - 3 + 2\sqrt{3} \).Adding results: \[ (\frac{\pi}{3} - 1) + (\frac{2\pi}{3} - 3 + 2\sqrt{3}) = \pi - 4 + 2\sqrt{3} \]

Step 6: Choose the Correct Option

Match the calculated result \( \pi - 4 + 2\sqrt{3} \) with the given options. The closest option is (C) \(4 \sqrt{3}-4\) but notice the proper evaluation does not match directly, hence reconsider exact matches or typographical errors in results.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a clever method used in calculus to simplify integrals involving square roots. It does this by transforming the integral into an easier form using trigonometric identities. This substitution often helps to manage complex expressions that include squares, as seen in the original exercise's \( \sqrt{1+4 \sin ^{2} \frac{x}{2}-4 \sin \frac{x}{2}} \).
For this problem, start by using the identity \( \sin^2 \frac{x}{2} = 1 - \cos^2 \frac{x}{2} \).This substitution simplifies the expression by eliminating squares and makes the expression under the square root easier to handle.

• This method is beneficial because it turns complex square roots into absolute values.
• It leverages trigonometric identities to transform the expression into a more calculable form.

In our exercise, using trigonometric substitution was the key to transform the integral into a manageable solution.

Definite Integrals

Definite integrals are a key concept in calculus, fundamentally representing the area under a curve between two points. For the integral \( \int_{0}^{\pi} \sqrt{1+4 \sin ^{2} \frac{x}{2}-4 \sin \frac{x}{2}} \, dx \), the bounds are from \( 0 \) to \( \pi \),indicating the interval over which we're calculating this area.

In this exercise, the definite integral helps us calculate a finite area, even though the expression is complex due to the square root and trigonometric functions. Key aspects of calculating definite integrals include:

• Applying limit boundaries to the function.
• Properly evaluating the integral over the given interval.

Breaking down the integral, especially when involving a point where the function changes (as with the absolute value integration), is crucial for an accurate result.

Absolute Value Integration

Absolute value integration deals with situations where the expression within an integral might change sign over the interval. This is essential when dealing with expressions like \(|1 - 2\sin \frac{x}{2}|\) in our exercise.

The expression \(1 - 2\sin \frac{x}{2}\) becomes zero at \(x = \frac{\pi}{3}\),splitting the integral into two distinct parts to handle the change in the sign of the function smoothly:

• From \( 0 \) to \( \frac{\pi}{3} \), the expression \(2\sin \frac{x}{2} - 1\) is used.
• From \( \frac{\pi}{3} \) to \( \pi \), the expression \(1 - 2\sin \frac{x}{2}\) is used.

Using absolute value integration ensures that we accurately capture the total area, taking into account how the function's behavior changes across different intervals. This approach is crucial for ensuring the computed definite integral accounts for all positive and negative areas under the curve.