Problem 54 - Calculus: Early Transcendental Functions

Question

Generalize the result of exercise 53 to \(\int_{2}^{4} \frac{f(9-x)}{f(9-x)+f(x+3)} d x\) for any positive, continuous function \(f\) on [2,4]

Step-by-Step Solution

Answer

The generalized solution to the given integral is 2.

Step 1: Analyze the given

The exercise provides the integral \(\int_{2}^{4} \frac{f(9-x)}{f(9-x)+f(x+3)} dx\).

Step 2: Use transformation property

By using the transformation property of integration, let \(u = 9-x\), then \(du = -dx\) and when \(x = 2, u = 7\) and when \(x = 4, u = 5\). Change the limits of integration and the integrand appropriately. Therefore, the transformed integral is \(-\int_{7}^{5}\frac{f(u)}{f(u)+f(12-u)} du\)

Step 3: Change the order of integration

It's known that changing the order of integration gives the negative of the integral. Therefore, \(-\int_{7}^{5}\frac{f(u)}{f(u)+f(12-u)}du = \int_{5}^{7}\frac{f(u)}{f(u)+f(12-u)} du\).

Step 4: Averaging the integrals

Now, add the transformed integral to the original integral and then divide by 2. We get, \(\frac{1}{2}[\int_{5}^{7}\frac{f(u)}{f(u)+f(12-u)} du + \int_{2}^{4}\frac{f(9-x)}{f(9-x)+f(x+3)} dx] = \frac{1}{2}[ \int_{5}^{7} du + \int_{2}^{4} dx] = \frac{1}{2}[(7-5) + (4-2)]\)

Step 5: Solve the integral

Now, we solve the above integral, which yields \(\frac{1}{2}[(7-5) + (4-2)] = \frac{1}{2}*4 = 2\)

Key Concepts

Transformation Property

The transformation property is a basic yet powerful tool in the realm of integration techniques. It allows us to simplify complex integrals by changing variables. This is often done when the function or its bounds make direct integration challenging.

In this specific exercise, we use the transformation property by setting a new variable, replacing the existing one to simplify the expression. We choose a substitution, here shown as \( u = 9-x \). When you substitute \( u \, \text{for} \, x \), remember that this changes the differential \( dx \) to \( du \), also flipping the sign, \( du = -dx \). This is why the integral signs get reversed, creating a negative, or flipped, integral.

• First, identify complicating parts in the integrand.
• Introduce a new variable that simplifies these parts.
• Change all occurrences of the old variable to the new variable, including transformed limits.

By changing the bounds from \([2,4]\) for \( x \) to \([7,5]\), which becomes \([-\int_{7}^{5}...]\), the transformation property helps to unfold symmetries or simplify the integral's evaluation.

Order of Integration

Understanding the order of integration is key to manipulating integrals effectively. Once you've changed the variable and possibly reversed the integral as outlined in the transformation property, you may find that flipping it back to a positive orientation leads to a solution. This is done by altering the limits of integration once more.

Let's say you had already adjusted your limits using the transformation property, converting \([-\int_{7}^{5}...]\) into \(\int_{5}^{7}...\). By reversing the limits, we remove the negative sign from the integral, simplifying the solving process.

• Acknowledge that reversing limits multiplies the integral by \(-1\).
• Choose an orientation that best suits further calculations or interpretations.
• Transform the integral back to positive if required by adjusting the bounds again.

This technique is handy, ensuring no extra negatives linger to complicate finding the integral’s result.

Averaging Integrals

Averaging integrals, especially in the context of symmetry, simplifies the problem substantially. It operates on the principle that if you add two integrals, one being the transformation of the other, their average can often lead to simpler bounds and calculations.

Consider you have two integrals, one from the original function and one from its variable substitution. By averaging them, we effectively create uniformity in the problem, making solutions like \(\int_{5}^{7}...\) and \(\int_{2}^{4}...\) easier to handle. The resulting integral is easier to evaluate since symmetries often cancel out more complex behaviors within the function.

• Take both the original and transformed integrals.
• Combine and divide by 2, effectively averaging them.
• Solve the simplified expression yielding constant bounds or results.

In this exercise, averaging leads to constant limits when simplified—\((2 + 2)/2\), essentially reducing to a simple value or structure to further elucidate f’s symmetrical behaviors over the interval considered.