Problem 206 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int_{0}^{\pi} x f(\sin x) d x\) equal to (A) \(\pi \int_{0}^{\pi} f(\cos x) d x\) (B) \(\pi \int_{0}^{\pi}(\pi-x) f(\sin x) d x\) (C) \(\frac{\pi}{2} \int_{0}^{\pi / 2} f(\sin x) d x f g p\) (D) \(\pi \int_{0}^{\pi / 2} f(\cos x) d x\)

Step-by-Step Solution

Answer

The answer is (B).

Step 1: Identify the Integral Transformation

The given integral is \( \int_{0}^{\pi} x f(\sin x) \, dx \). We need to examine if a change of variables can simplify this integral for easier comparison to the options provided.

Step 2: Apply Substitution

Consider the substitution \( u = \pi - x \). Then, \( du = -dx \), and the limits change from \( x = 0 \) to \( u = \pi \) and \( x = \pi \) to \( u = 0 \). Reversing the limits yields: \[ \int_{0}^{\pi} x f(\sin x) \, dx = \int_{\pi}^{0} (\pi - u) f(\sin(\pi - u)) (-1) \, du. \]

Step 3: Simplify Using Trigonometric Identities

Since \( \sin(\pi - u) = \sin u \), the integral becomes: \[ \int_{0}^{\pi} (\pi - u) f(\sin u) \, du, \] which matches option (B) exactly.

Step 4: Verify the Solution

Comparing the transformed integral \( \int_{0}^{\pi} (\pi - x) f(\sin x) \, dx \) with the options given, this expression clearly aligns with option (B).

Key Concepts

Integral Transformation

Integral transformation is a crucial technique in calculus. It enables us to alter the form of an integral, making it easier to evaluate or compare against a given result. This process often involves substituting variables or adjusting limits of integration.
In our exercise, the integral transformation is evident from the start, as we aim to simplify the expression \( \int_{0}^{\pi} x f(\sin x) \, dx \). By appropriately transforming the integral, we align it with one of the offered solutions. The transformation typically involves analytical strategies and systematic approaches.

• Change of variable: A common method to simplify an integral.
• Adjustment of limits: Important to maintain the integrity of the integral.
• Comparison with results: After transformation, compare the new integral form to provided options.

This transformation not only facilitates solving the approach efficiently but also enhances understanding of integral properties.

Trigonometric Identities

Trigonometric identities are foundational when dealing with integrals involving trigonometric functions. They offer simplifications which make the integration process more manageable.
In the step by step solution of our problem, we encounter the identity \( \sin(\pi - u) = \sin u \). This identity is crucial, as it allows the transformation of the integral without altering its value. Proper utilization of trigonometric identities here is key in solving the exercise.

• They help transform complex expressions into simpler forms.
• Preserve the essence of trigonometric functions.
• Assist in integrating functions over specific limits.

By incorporating these identities, we transform an originally daunting integral into a more tractable form. This understanding is critical not only to solve this exercise but also to handle similar future problems.

Substitution Method

The substitution method in calculus is an invaluable tool that simplifies the integration process. It involves changing variables to make an integral easier to solve.
In our exercise, we employed the substitution \( u = \pi - x \). With this, the differential changes to \( du = -dx \), and the limits transform correspondingly. By reversing the limits and sign, we successfully reformulate the integral.

• Substitution simplifies intricate integrals to familiar forms.
• Limits of integration: Upon substituting, these may need adjustment, ensuring they correspond to the new variable.
• Integral reversal: Makes the integration process align with mathematical conventions.

The substitution method bridges the complex and the straightforward, easing the path to solution. It's a powerful strategy that enhances problem-solving capabilities, as witnessed in our example, directing us effortlessly towards the correct option.