Problem 261 - Calculus Volume 1

Question

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(x+1)^{4} d x ; u=x+1 $$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{(x+1)^5}{5} + C \).

Step 1: Identify the substitution

We are given the substitution \( u = x + 1 \). This means whenever we encounter \( x + 1 \) in the expression, we can replace it with \( u \).

Step 2: Derive the differentials

Differentiate \( u = x + 1 \) with respect to \( x \), to find \( \frac{du}{dx} = 1 \). Hence, \( du = dx \), allowing us to replace \( dx \) with \( du \) in the integral.

Step 3: Substitute and rewrite the integral

Replace \( x + 1 \) with \( u \) and \( dx \) with \( du \) in the integral. This changes the integral from \( \int (x + 1)^4 \, dx \) to \( \int u^4 \, du \).

Step 4: Integrate with respect to u

Find the antiderivative of \( u^4 \). The antiderivative of \( u^n \) is \( \frac{u^{n+1}}{n+1} + C \). Thus, the antiderivative of \( u^4 \) is \( \frac{u^5}{5} + C \).

Step 5: Substitute back to x terms

Replace \( u \) with \( x + 1 \) to revert back to the original variable. The antiderivative becomes \( \frac{(x+1)^5}{5} + C \).

Step 6: Simplify, if necessary

In this case, the expression \( \frac{(x+1)^5}{5} + C \) is already simplified. Thus, there is nothing left to simplify further.

Key Concepts

Integration by Substitution

Integration by substitution is a highly useful technique when it comes to finding antiderivatives, particularly in integrals where a part of the function can be represented as another variable. The main idea of substitution is to simplify the given integral by defining a new variable. This method is similar to the reverse mechanism of the chain rule used in differentiation.

In essence, substitution involves a few key steps:

• Select a substitution variable, often denoted as \( u \), that simplifies the function.
• Recalculate the differential, substituting variables in the integrand.
• Transform the integral into a simpler form, integrate, and then substitute back the original variables.

In our original exercise, choosing \( u = x + 1 \) and finding \( du = dx \) simplified the integral \( \int (x + 1)^4 \, dx \) to \( \int u^4 \, du \). This simplified form makes it much easier to integrate, as we just integrate \( u^4 \) with respect to \( u \). Remember, choosing the right substitution is pivotal for successfully applying this method.

Definite Integral

Understanding a definite integral requires delving into integrating a function over a specific interval \([a, b]\). This type of integral yields a precise numerical value and is used to find areas under curves, among other applications. When evaluating definite integrals, the limits change according to the substitution.

In integration by substitution, when dealing with a definite integral, one must:

• Convert the limits of integration according to the substitution \( u = g(x) \).
• Substitute and simplify the integral as you would normally do in substitution methods.
• Calculate the new definite integral over the interval \([g(a), g(b)]\).
• There's no need to change back to the original variable after integration as long as the limits have been converted.

Keep in mind that definite integrals are not necessarily involved in every solution but understanding their role enhances the comprehension of integration processes and applications.

Indefinite Integral

Indefinite integrals represent a family of functions and are denoted without bounds, typically yielding a generic form of an antiderivative plus a constant \( C \). This constant denotes an unlimited set of possible solutions because differentiation eliminates constants. Unlike definite integrals, indefinite integrals do not have a specific boundary, and the result, therefore, is not a specific value but a function.

In practical terms, when performing integration:

• An indefinite integral can be thought of as the inverse operation of differentiation.
• Results include \( + C \), which accounts for any shift in the vertical position of the antiderivative.
• The process consists of finding a general formula for the antiderivative of a function.

In our original exercise, the solution led to the antiderivative \( \frac{(x+1)^5}{5} + C \). This is an indefinite integral because it supports all possible vertical shifts of the curve described by this formula. Therefore, the infinite "family" of antiderivatives visually illustrates these many possible solution branches.