Problem 46 - Calculus

Question

Prove (by a substitution) that $$ \int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x $$

Step-by-Step Solution

Answer

The substitution \( u = -x \) proves the integral equality by changing limits and factoring \( -du \).

Step 1: Understanding the Problem

We need to prove that \( \int_{a}^{b} f(-x) \, dx = \int_{-b}^{-a} f(x) \, dx \). This involves showing that both integrals are equal using a substitution method.

Step 2: Choosing a Substitution

We can use a substitution to transform the integrand of the left side integral. Let's choose \( u = -x \). This substitution will help us convert \( f(-x) \) on the left side to an equivalent form on the right.

Step 3: Applying the Substitution

Using the substitution \( u = -x \), compute the differential: \( du = -dx \) or \( dx = -du \). We also need to adjust the limits of integration; when \( x = a, \, u = -a \), and when \( x = b, \, u = -b \).

Step 4: Changing Limits and Integrand

Substitute into the integral \( \int_{a}^{b} f(-x) \, dx \) to get \( \int_{-a}^{-b} f(u) (-du) \). The negative sign can be factored out, changing the limits to \( \int_{-b}^{-a} f(u) \, du \). This matches the target integral \( \int_{-b}^{-a} f(x) \, dx \) by simply relabeling \( u \) as \( x \).

Step 5: Final Verification

After relabeling \( u \) back to \( x \), we observe that both sides of the equation are indeed equivalent after substitution. Hence, the integral equality \( \int_{a}^{b} f(-x) \, dx = \int_{-b}^{-a} f(x) \, dx \) holds true.

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify the process of finding an integral. It's often compared to reversing the chain rule from differentiation. To apply it effectively:

• Choose a suitable substitution, often denoted as \( u \), which simplifies the integrand.
• Calculate the differential, \( du \), in terms of the original variable, \( x \).
• Replace the integrand and differential in the integral, converting the problem into a simpler form.

In our exercise, we used the substitution \( u = -x \) to transform the integrand \( f(-x) \). This substitution simplifies the evaluation by converting the original problem into a familiar integral format.
By systematically changing variables, integration by substitution can make seemingly complex integrals more manageable.

Limits of Integration

When performing integration by substitution, the limits of integration, which define the bounds of the integral, must also be changed to reflect the substitution. This involves:

• Identifying the new limits by substituting the original boundary values into the substitution equation.
• Keeping track of these changes meticulously, as they can fundamentally alter the integral.

In the problem at hand, the original limits \( a \) and \( b \) are transformed. For our substitution \( u = -x \), when \( x = a \), \( u = -a \), and when \( x = b \), \( u = -b \).

These new limits ensure that the transformed integral is correctly defined within the changed variable's context.

Change of Variables

The change of variables is a pivotal step in integration by substitution, used to simplify the integrand and compute the integral more easily. It works by transforming the variable of integration to a new variable.

• This simplifies the function inside the integral, making it easier to evaluate.
• Ensures that the integral retains its original area or value.

In our exercise, changing from \( x \) to \( u \) using \( u = -x \) allowed the troublesome \( f(-x) \) to be rewritten as \( f(u) \). This transformation helps in aligning the integral with the required form so that it can be simplified and manipulated easily.

The essential idea is that this procedure rephrases the problem, maintaining equivalence between the original and substituted integrals.

Integral Equality Proof

An integral equality proof demonstrates that two integrals are equal. It often involves using substitutions, as illustrated in our exercise. Here is how:

• Apply substitution to change the integrand and limits of the first integral.
• Relabel variables to match the form of the second integral.
Provide a clear and logical explanation, verifying each step to ensure accuracy.

In our exercise, the proof involved verifying that by substituting \( u = -x \), and accordingly changing limits, the integral \( \int_{a}^{b} f(-x) \, dx \) becomes \( \int_{-b}^{-a} f(u) \, du \). By relabeling \( u \) as \( x \), we end up with the target integral \( \int_{-b}^{-a} f(x) \, dx \).

Thus, we proved that the two integrals are indeed equal, confirming that the integrals over two different intervals can have the same value when appropriately manipulated.