Problem 7 - Calculus

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=1 / \sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = 3x^{1/3} + C \).

Step 1: Rewrite the expression in power form

The given function is \( f(x) = \frac{1}{\sqrt[3]{x^2}} \). Rewrite it in terms of a power by recognizing that \( \sqrt[3]{x^2} \) is equivalent to \( x^{2/3} \). Thus, \( \frac{1}{x^{2/3}} = x^{-2/3} \). The function now is \( f(x) = x^{-2/3} \).

Step 2: Apply antiderivative power rule

Use the power rule for integration, which states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), provided \( n eq -1 \). Here, \( n = -\frac{2}{3} \), so add 1 to the exponent and divide by the new exponent: \( F(x) = \frac{x^{(-2/3)+1}}{(-2/3)+1} = \frac{x^{1/3}}{1/3} \).

Step 3: Simplify the antiderivative

Simplify \( \frac{x^{1/3}}{1/3} \) by multiplying the numerator by the reciprocal of \( \frac{1}{3} \). This results in \( 3x^{1/3} \).

Step 4: Add the constant of integration

The general antiderivative includes an arbitrary constant \( C \). Therefore, the general antiderivative of \( f(x) = x^{-2/3} \) is \( F(x) = 3x^{1/3} + C \).

Key Concepts

Power Rule

The power rule is a key part of finding antiderivatives of functions. It helps simplify the process of integration. This rule is especially useful when dealing with polynomial expressions or terms expressed in powers. The power rule for integration states that the antiderivative of a function of the form \(x^n\) is \(\frac{x^{n+1}}{n+1}\), as long as \(n eq -1\). This rule essentially "reverses" the process of differentiation by putting back the power of the term and adjusting its coefficient. To apply the power rule, you first add 1 to the exponent \(n\) in the term \(x^n\). Then, you divide the resulting expression by the new exponent. This gives you the antiderivative for the expression. Keep in mind, if your function is not in the form of \(x^n\), you may need to rewrite it first, as shown in this exercise, where \(1/\sqrt[3]{x^2}\) was rewritten as \(x^{-2/3}\). Once in this form, applying the power rule becomes straightforward.

Constant of Integration

When you calculate an antiderivative, don't forget about the constant of integration, often denoted by \(C\). This constant is crucial because it represents all possible "shifts" of the antiderivative function along the y-axis. Every indefinite integral has a whole family of curves that can be solutions, which is why we add \(C\). Each different value for \(C\) corresponds to a different member of this family.The constant of integration arises because when taking the derivative of a constant, the result is zero. Therefore, without additional information such as an initial condition, the specific value of \(C\) remains indefinite. In the context of this exercise, once you find \(F(x) = 3x^{1/3}\), by adding \(C\), you make it the most general form of the antiderivative.

Integration Techniques

Integration techniques are methods employed to find antiderivatives and definite integrals. Often functions will not be presented in a straightforward \(x^n\) format, and understanding various techniques dynamically equips you to handle such challenges.

• Substitution Method: Look for an internal function and its derivative, commonly used for integrals involving composite functions.
• Integration by Parts: Useful when the product of two functions is involved, providing a way to rearrange and solve complex integrals.
• Rewriting Expressions: Like in this exercise, sometimes just rewriting the function helps. By expressing \(\frac{1}{\sqrt[3]{x^2}}\) as \(x^{-2/3}\), it became a straightforward application of the power rule.

Learning and practicing these techniques can make integration much easier and more intuitive, even for seemingly complex problems.