Problem 4 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\)

Step-by-Step Solution

Answer

The identity is confirmed as true after substituting and transforming the integral.

Step 1: Understanding the Problem

We are given an integral identity to solve: \(\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\). Our goal is to understand this identity and verify the equation is true.

Step 2: Define a Substitution Variable

To solve the problem, let us use a substitution. Define a new variable \(u = a + b - x\), aimed at transforming the integral. This substitution will help match the form of the second integral.

Step 3: Calculate Differential

Find the differential of \(u\): \(du = -dx\). This will be used to replace \(dx\) in the original integral.

Step 4: Change the Limits of Integration

When \(x = a\), \(u = a + b - a = b\). When \(x = b\), \(u = a + b - b = a\). Thus, the limits of integration change from \([a, b]\) to \([b, a]\).

Step 5: Substitute and Transform the Integral

Substitute \(x\) with \(a+b-u\) and \(dx\) with \(-du\) in the original integral, changing the limits from \([a, b]\) to \([b, a]\): \[\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(u) du = \int_{a}^{b} f(u) du\]. Notice the negative sign flips the limits back to \([a, b]\), and we get \(\int_{a}^{b} f(a+b-x) dx\).

Step 6: Verify the Identity

After substitution and transformation, we have shown that the integral \(\int_{a}^{b} f(x) dx\) becomes \(\int_{a}^{b} f(u) du\), where \( u = a+b-x\). This proves that the given identity \(\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\) holds true.

Key Concepts

Substitution Method

The substitution method is a technique often used in calculus to simplify integrals. It works by changing the variable of integration to make the integration process easier. By substituting a new variable for a complex expression, the integral's structure often becomes more straightforward.Here, the substitution method is used to verify the given integral identity. We substitute the variable \(x\) with a new variable \(u\) defined by the equation \(u = a + b - x\). This substitution aims to transform the integral into a form that closely resembles the other side of the identity.

• By substituting \(u\), complex expressions often simplify.
• Adjusting the differential \(du = -dx\) aligns the differentials appropriately.
• Altering the limits of integration ensures we integrate over the correct range.

This method not only aids in solving the problem but also deepens understanding of symmetry in integrals.

Integral Identity

An integral identity is a statement asserting that two integrals are equal across a specified interval. These identities often reveal deeper properties of functions and their mirror-like transformations.In this scenario, the identity claims: \[\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\]Examining each side through transformation, we grasp how the functions relate through their mirroring across the line \(x = \frac{a+b}{2}\).

• Integral identities demonstrate symmetries in functions.
• They allow the interchange of variable transformations freely.
• Such identities are useful for verifying and solving integrals efficiently.

Understanding integral identities like this one emphasizes the flexibility and power of integrals in calculus.

Limits of Integration

The limits of integration define the range over which the integral is evaluated. These bounds are crucial when exchanging variables and applying substitutions.Initially, the integration is over the interval \([a, b]\). With the substitution \(u = a + b - x\), the limits switch:- When \(x = a\), \(u = b\).- When \(x = b\), \(u = a\).This change emphasizes the significance of direction and limits in defining an integral:

• Switching limits inverses the integration direction, adding a negative sign.
• Returning limits to \([a, b]\) using \(-du\) confirms integral properties.
• Mastery of manipulating these boundaries enhances problem-solving skills.

Changing the limits in line with variable swaps ensures the integrity and correctness of the integral transformations.