Problem 96 - Essential Calculus

Question

Let \(\int_{-\infty}^{\infty} f(x) d x\) be convergent and let \(a\) and \(b\) be real numbers where \(a \neq b\). Show that \(\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x\)

Step-by-Step Solution

Answer

The statement \(\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x\) is proven to be true, given the conditions that \(\int_{-\infty}^{\infty} f(x) d x\) is convergent and \(a \neq b\), utilizing the property of convergent integrals.

Step 1: Identify the given conditions

The given conditions of this problem are $\int_{-\infty}^{\infty} f(x) d x$ is convergent and $a \neq b$ are real numbers.

Step 2: Refer to the Property of Convergent Integrals

Since the integral from \(-\infty\) to \(+\infty\) is convergent, it can be split at any point \(x = c\). Therefore, the integral \(\int_{-\infty}^{\infty} f(x) d x\) can be written as \(\int_{-\infty}^{c} f(x) d x + \int_{c}^{\infty} f(x) d x\). It is essential to note that this is true for any real value \(c\).

Step 3: Apply Property to Given Integrals

As per step 2, it can be shown that: \(\int_{-\infty}^{a} f(x) d x + \int_{a}^{\infty} f(x) d x = \int_{-\infty}^{\infty} f(x) d x\) and \(\int_{-\infty}^{b} f(x) d x + \int_{b}^{\infty} f(x) d x = \int_{-\infty}^{\infty} f(x) d x\)

Step 4: Show Equality

From step 3, as both sides of the equation equal \(\int_{-\infty}^{\infty} f(x) d x\), it can hence be shown that they equal one another, as such: \(\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x\). This completes the proof.

Key Concepts

Convergence of integrals

When dealing with improper integrals, one crucial aspect to understand is convergence. An integral converges when it approaches a finite limit as the integration domain extends to infinity. Specifically, for \( \int_{-\infty}^{\infty} f(x) \, dx \) to converge, both integrals \( \int_{-\infty}^{c} f(x) \, dx \) and \( \int_{c}^{\infty} f(x) \, dx \) need to reach finite values regardless of the partition point \(c\).
Convergence is essential because it guarantees the integral has a concrete value, similar to finding the area under a curve. To check for convergence, analyze the limits of the function and ensure they do not diverge as they approach infinity.

• Convergence involves evaluating whether an integral results in a finite number.
• Improper integrals are split into two parts, each approaching infinity.
• Both parts need to converge separately for the entire integral to converge.

In our problem, the convergence of \( \int_{-\infty}^{\infty} f(x) \, dx \) allows us to split at any point \(a\) or \(b\) and still reach the same total.

Properties of integrals

The properties of integrals, particularly concerning improper integrals, allow us to manipulate and evaluate these expressions effectively. One fundamental property is linearity, meaning the integral of a sum is the sum of the integrals. This property is used to split integrals into more manageable parts.
For example, if an integral \( \int_{-\infty}^{\infty} f(x) \, dx \) is solvable, it means we can determine individual sections, like \( \int_{-\infty}^{a} f(x) \, dx \) and \( \int_{a}^{\infty} f(x) \, dx \) for any point \(a\). Another property of convergent integrals is that they remain unchanged when the interval is reshuffled, showing that: \[ \int_{-\infty}^{a} f(x) \, dx + \int_{a}^{\infty} f(x) \, dx = \int_{-\infty}^{b} f(x) \, dx + \int_{b}^{\infty} f(x) \, dx \]

• Linearity lets us separate integral calculations, which is especially useful when dealing with infinity.
• This problem uses linearity to showcase that integrals can be split or combined without changing the result.
• Remember: Improper integrals can be divided at any point without affecting the convergence.

By understanding these properties, you can confidently tackle complex integral problems with multiple parts.

Splitting the integral

Splitting the integral is a technique used to simplify and focus on specific segments of an integral. In the context of improper integrals, it involves dividing the integral into two manageable parts across a specified point, which can be any real number.
In the exercise, we see this splitting process happen when it states: \[ \int_{-\infty}^{a} f(x) \, dx + \int_{a}^{\infty} f(x) \, dx = \int_{-\infty}^{b} f(x) \, dx + \int_{b}^{\infty} f(x) \, dx \] Each side of the equation splits the old interval of \((-\infty, \infty)\) into parts before and after the point \(a\) or \(b\). By specifying alternative split points \(a\) and \(b\), students can see how the integral's total remains constant due to convergence.

• Splitting helps in breaking down complex problems into solvable parts.
• Any real number can serve as a split point without affecting the integral's total.
• This is highly advantageous when tackling integrals across entire real lines.

By mastering splitting, you'll have an essential tool for handling more challenging integrals, making complex calculations more straightforward.