Problem 48 - Calculus Single Variable

Question

Use the Chain Rule to calculate the given indefinite integral. $$ \int x\left(x^{2}+1\right)^{100} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{202} (x^2 + 1)^{101} + C \).

Step 1: Identify Inner Function

First, recognize the inner function in the expression. Here, the inner function is \( u = x^2 + 1 \).

Step 2: Differentiate Inner Function

Differentiate the inner function with respect to \( x \), giving \( \frac{du}{dx} = 2x \).

Step 3: Substitute and Express in Terms of \( u \)

Rewrite the integral using the inner function \( u \). Substitute \( u = x^2 + 1 \) and \( du = 2x \ dx \), which implies that \( x \ dx = \frac{1}{2} du \). The integral becomes \( \frac{1}{2} \int u^{100} \, du \).

Step 4: Integrate with Respect to \( u \)

Perform the integration with respect to \( u \): \( \int u^{100} \, du = \frac{u^{101}}{101} + C \). Multiply by \( \frac{1}{2} \): \( \frac{1}{2} \left( \frac{u^{101}}{101} \right) + C = \frac{1}{202} u^{101} + C \).

Step 5: Substitute Back the Inner Function

Replace \( u \) with the original expression \( x^2 + 1 \). Thus, the solution becomes \( \frac{1}{202} (x^2 + 1)^{101} + C \).

Key Concepts

Understanding the Chain Rule

The Chain Rule is a crucial concept in calculus, especially when tackling problems involving differentiation and integration. It helps us take derivatives of composite functions, which are functions made up of one function inside another.
Imagine you have an outer function, usually denoted as \( f(u) \), and an inner function, denoted as \( u(x) \). The Chain Rule states that the derivative of \( f(u(x)) \) is found by multiplying the derivative of the outer function by the derivative of the inner function. Mathematically, it's expressed as: \[ \frac{d}{dx}[f(u(x))] = f'(u(x)) \, \cdot \, u'(x) \]
In the context of integration, the Chain Rule's cousin is used, often called "Integration by Substitution". This method simplifies the integral by substituting the inner function, allowing for easier computation. Understanding this rule is essential for solving integrals like the one in the exercise.

The Indefinite Integral Simplified

An indefinite integral represents a family of functions with an added constant. It's essentially the reverse process of differentiation. Instead of finding the rate of change, we determine the accumulated quantity whose derivative gives us the integrand.
In mathematical notation, the indefinite integral of a function \( f(x) \) is written as \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative, and \( C \) is the constant of integration.
The indefinite integral is called so because it involves an unknown constant, unlike definite integrals, which calculate a specific value over an interval. When tackling more complex integrations, like the one in our exercise, we often use substitution to transform it into a simpler form. This process involves recognizing patterns and applying substitution correctly, in order to find the antiderivative effectively.

Deciphering the Inner Function

The inner function plays a pivotal role in problems involving the Chain Rule or integration by substitution. It is the function inside another function - the building block of composite functions. In our exercise, the inner function is \( u = x^2 + 1 \).
Identifying the inner function correctly is the first step in solving integrals by substitution. By substituting the inner function as a single variable (often \( u \)), you transform a complicated problem into a simpler one. This transformation allows you to deal with the integral in terms of just one variable.
Once the inner function is substituted, and after integrating, it's essential to revert back to the original variable. This ensures that the final result of the integration is expressed in terms of the initial variable \( x \), maintaining the integrity of the original problem.

The Role of Differentiation in Integration by Substitution

Differentiation is the process of finding the derivative of a function, revealing the rate at which a function changes. In integration by substitution, differentiation assists us in transforming variables seamlessly.
When working with integration by substitution, you begin by differentiating the inner function. This derivative, \( \frac{du}{dx} \), helps to reconfigure the differential \( dx \) in terms of \( du \). Essentially, you're setting up a bridge between the original variable and the substituted variable.
For example, in our exercise, the inner function \( u = x^2 + 1 \) is differentiated to produce \( \frac{du}{dx} = 2x \). This derivative ensures that the corresponding \( dx \) is replaced properly, in this case, setting \( x \, dx = \frac{1}{2} du \). Properly managing this step is crucial for successful integration and finding the correct antiderivative.