Problem 74 - Applied Calculus

Question

We omit the constant of integration when we integrate \(d v\) to get \(v\). Including the constant \(C\) in this step simply replaces \(v\) by \(v+C,\) giving the formula $$ \int u d v=u(v+C)-\int(v+C) d u $$ Multiplying out the parentheses and expanding the last integral into two gives $$ \int u d v=u v+C u-\int v d u-C \int d u $$ Show that the second and fourth terms on the right cancel, giving the "old" integration by parts formula \(\int u d v=u v-\int v d u .\) This shows that including the constant in the \(d v\) to \(v\) step gives the same formula. One constant of integration at the end is enough.

Step-by-Step Solution

Answer

The constant terms cancel out, giving the formula \(\int u dv = uv - \int v du\).

Step 1: Understanding the Problem

We need to verify if the constant of integration needed when integrating \(dv\) into \(v\) can be omitted without affecting the final integration by parts formula. We start with the given expression \(\int u \, dv = u(v+C) - \int (v+C) \, du\) and apply expansion.

Step 2: Expanding the Expression

The expression \(\int u \, dv = u(v+C) - \int (v+C) \, du\) simplifies to \(\int u \, dv = uv + Cu - \int v \, du - C \int du\) by distributing terms inside the integral \(\int (v+C) \, du\) and individually integrating \(v\) and \(C\).

Step 3: Canceling Constant Terms

Notice that the term \(Cu\) gets canceled by the term \(-C \int du\) because \(\int du = u + C'\), where \(C'\) is another constant, which maintains \(Cu\) as effectively constant in the context of definite integrals.

Step 4: Conclusion with Original Formula

With \(Cu\) and \(-C \int du\) canceling each other out, we are left with \(\int u \, dv = uv - \int v \, du\). This is the original integration by parts formula, confirming that the constant \(C\) does not affect the integral result when omitted.

Key Concepts

Constant of Integration

When integrating a function, a constant of integration is an arbitrary constant added to the result of the indefinite integral. This constant, often denoted as \(+ C\), represents any possible constant term since the derivative of a constant is zero.

For instance, when you integrate a simple function \(f(x) = 2x\), the indefinite integral becomes \(F(x) = x^2 + C\). The \(+ C\) represents that there are infinitely many functions with the same derivative \(2x\), each differing by a constant. This is the reason why every indefinite integral has this constant.

In the context of integration by parts, when integrating \(dv\), adding \(+ C\) doesn't change the operation since this constant specifically cancels out when dealing with functional expressions and definite boundaries. Therefore, in practice, some steps omit this constant in equations to keep expressions straightforward. A single constant of integration at the end of calculations is usually sufficient to account for all potential constants introduced throughout the integration process. This avoids excessively complicated expressions.

Definite Integral

A definite integral is an integral with upper and lower limits, typically used to calculate the area under a curve between two points. These limits are numbers that define the range of integration, and they make the result a specific numerical value rather than a general function.

The process of evaluating a definite integral involves finding the antiderivative, just like with indefinite integrals. However, after obtaining the antiderivative, you subtract the value of this function at the lower limit from its value at the upper limit. This is mathematically represented as:

• \(\int_a^b f(x) \, dx = F(b) - F(a)\)

The resulting value from this operation gives the net "area" between the curve \(f(x)\) and the x-axis from \(x = a\) to \(x = b\). It's important to note that the constant of integration \(C\) does not appear in definite integrals because when calculating the difference, the constants cancel each other out.

Indefinite Integral

An indefinite integral, unlike a definite integral, does not have limits. It represents a family of functions, and its outcome is a general antiderivative requiring \(+ C\), the constant of integration.

Mathematically, it is denoted by:

• \(\int f(x) \, dx = F(x) + C\)

Here, \(F(x)\) is the antiderivative of \(f(x)\), and \(+ C\) indicates that there are infinitely many antiderivatives, each differing by a constant.

In the context of integration by parts, the indefinite integral connects closely to how we derive functions and formulas involving variables and constants. This concept is pivotal because it allows analysts and mathematicians to work with flexible solutions that can vary with additional constraints or modifications. Understanding indefinite integrals forms the foundation for topics such as differential equations and further applications of calculus.