Problem 18 - Calculus for Biology and Medicine

Question

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int(4+x)^{1 / 7} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{7}{8} (4 + x)^{8/7} + C \).

Step 1: Identify the substitution

We decide to use the substitution method. Since we have the integral \( \int(4+x)^{1 / 7} \, dx \), we substitute \( u = 4 + x \). We then differentiate this substitution to find \( du \).

Step 2: Differentiate the substitution

From the substitution \( u = 4 + x \), we differentiate to get \( \frac{du}{dx} = 1 \). Thus, \( du = dx \). This tells us that \( dx \) can be directly replaced by \( du \) in the integral.

Step 3: Rewrite the integral in terms of u

Replace \( 4 + x \) with \( u \) and \( dx \) with \( du \). The integral becomes \( \int u^{1/7} \, du \).

Step 4: Integrate with respect to u

Now integrate \( \int u^{1/7} \, du \) using the power rule for integration. The power rule states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Applying this rule: \( \int u^{1/7} \, du = \frac{u^{1/7 + 1}}{1/7 + 1} + C = \frac{u^{8/7}}{8/7} + C = \frac{7}{8} u^{8/7} + C \).

Step 5: Substitute back to x

Now substitute back using \( u = 4 + x \), hence the integral becomes \( \frac{7}{8} (4 + x)^{8/7} + C \). This is the indefinite integral of the given function.

Key Concepts

Substitution Method

The substitution method is a technique often used in integral calculus to simplify the process of finding an indefinite integral. The core idea is to replace a part of the integral with a single variable, usually denoted as \( u \). This method is particularly beneficial when dealing with composite functions that can otherwise appear complex and cumbersome to integrate directly by hand.
Here's how it generally works:

• You identify a suitable substitution \( u \), usually an expression you encounter repeatedly or a compound expression that simplifies the integral when isolated.
• Differentiate the substitution to express \( du \) in terms of \( dx \), rewriting the differential element in the integral accordingly.
• Substitute \( u \) and \( du \) back into the integral, transforming it into a simpler form.
• Once simplified, solve the integral in terms of \( u \). After calculating, substitute back the original variable expressions to return to the original terms of the integral.
  

By using substitution, you're effectively disguising a difficult integral as a much simpler one. The real magic is in properly choosing \( u \), which often determines the ease and success of this technique.

Indefinite Integrals

An indefinite integral is essentially the reverse process of differentiation. It represents a family of functions whose derivatives are known. When you see the integral sign \( \int \) without specific endpoints, you are dealing with an indefinite integral, which includes an arbitrary constant \( C \).
Let’s break it down:

• The goal is to find a function, or an antiderivative, whose derivative is the given function inside the integral.
• Indefinite integrals yield a general solution since you are not bound to any specific limits for evaluation.
• The constant \( C \) comes into play because the derivative of a constant is zero, thus the original function could've had any constant value added to it.

This concept lays the foundation for solving integrals without limits—a method crucial for many analytical processes in calculus. The integral provides a clear method to move backward from a derivative to the function itself.

Power Rule

The power rule is an essential technique in calculus used to find the integral of functions with powers of a variable. When it comes to indefinite integrals, the power rule simplifies the process by providing a straightforward formula. Here's how it unfolds:

• The power rule states that for any real number \( n \) (except \( n = -1 \)), the integral \( \int x^n \, dx \) is given by \( \frac{x^{n+1}}{n+1} + C \).
• This rule implies that you simply increase the exponent by 1 and divide by the new exponent, then add a constant \( C \).
• In the exercise, the integral \( \int u^{1/7} \, du \) is solved using the power rule, resulting in \( \frac{u^{8/7}}{8/7} + C \).

Understanding and applying the power rule can significantly ease the evaluation of integrals involving polynomials or powers, leading to quicker and more accurate solutions.