Problem 64 - Calculus: Early Transcendental Functions

Question

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int x^{2} \sqrt{x^{3}+2} d x$$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{2}{9} (x^{3}+2)^{3/2} + C \)

Step 1: Identify the function and its derivative

Look inside the square root and see if its derivative is among the other factors in the integral expression. Here notice \(x^{3} + 2\) is under the square root and its derivative \(3x^{2}\) is in the multiplication outside. However, instead of \(3x^{2}\) we have \(x^{2}\) so we introduce \(3\) and then we'll compensate by multiplying with \(1/3\) outside the integral. The integral becomes \( \frac{1}{3} \int 3x^{2} \sqrt{x^{3}+2} d x \)

Step 2: Substitute using the chain rule

Substitute \(u = x^{3} + 2. \, du = 3x^{2}dx \). The integral then becomes \( \frac{1}{3} \int \sqrt{u} du \)

Step 3: Evaluate Simplified Integral

The integral \( \frac{1}{3} \int \sqrt{u} du \) can be evaluated as \( \frac{1}{3} \cdot \frac{2}{3} u^{3/2} = \frac{2}{9} u^{3/2} \)

Step 4: Substitute back and simplify

Substitute \(u\) back into the equation to replace it with the original value of \(x^{3}+2\). The result is \( \frac{2}{9} (x^{3}+2)^{3/2} + C \), where \(C\) is the constant of integration.

Key Concepts

Chain Rule in Calculus

The chain rule is a fundamental tool in calculus for differentiating composite functions. When a function is the result of another function being applied to an inner function, like the layers of an onion, we use the chain rule to find its derivative.

For example, if we have a function expressed as \( h(x) = f(g(x)) \), the derivative of \( h(x) \) with respect to \( x \) is found by taking the derivative of \( f \) with respect to \( g \) and multiplying it by the derivative of \( g \) with respect to \( x \). This is expressed as \( h'(x) = f'(g(x)) \cdot g'(x) \).

In the context of the given exercise, the reverse process of the chain rule is applied during integration. We look for a function within the integral that has been composed with another function, and we make a substitution to simplify the integral into a form that is easier to integrate.

Integration Techniques

Integration is the process of finding the antiderivative of a function. It’s analogous to solving a puzzle with pieces that are different integration techniques, each suited to certain types of functions. Familiar techniques include substitution, integration by parts, partial fractions, and trigonometric substitutions, among others.

The appropriate choice of technique simplifies the integration process. Substitution, for example, is useful when a function contains another function times its derivative, as seen in the sample exercise. Introducing a new variable for the inner function and its differential can transform the integral into a more basic form, often leading to a solution involving standard integration formulas.

Improving Exercise Execution Using Integration Techniques

When applying integration techniques, it's key to recognize patterns and relationships among the functions involved. Understanding the connection between derivatives and integrals, as well as how different functions behave, can immensely improve one's ability to solve integrals.

Antiderivatives and Indefinite Integrals

An antiderivative of a function is another function that, when differentiated, yields the original function. Finding an antiderivative is akin to tracing back the steps of differentiation, and it’s represented by the indefinite integral. The process grants us the general form of antiderivatives, often denoted with a constant \(C\), symbolizing that there are infinitely many antiderivatives due to the constant term disappearing in differentiation.

The indefinite integral of a function \(f(x)\) is denoted as \(\int f(x) \, dx\), which represents the family of all antiderivatives of \(f(x)\). Understanding the underlying relationship between differentiation and integration is crucial for mastering the art of finding antiderivatives. The power rule for integration, which is the opposite of the power rule for differentiation, is often employed when integrating functions with powers, similar to the exercise provided.

Antiderivatives are more than just a mathematical process—they're a conceptual bridge connecting the rates of change to the accumulated quantity, essential for solving problems in physics, engineering, and economics.