Problem 281 - Calculus Volume 1

Question

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{3(x^3 - 3)} + C \).

Step 1: Choose a Suitable Substitution

To simplify the integral, let's try a substitution. Notice that the denominator's inside expression is complicating the integral. Choose the substitution \( u = x^3 - 3 \).

Step 2: Differentiate the Substitution

Differentiate both sides of the substitution \( u = x^3 - 3 \) with respect to \( x \) to find \( du \). This gives \( \frac{du}{dx} = 3x^2 \), or \( du = 3x^2 \, dx \).

Step 3: Solve for dx in Terms of du

From \( du = 3x^2 \, dx \), solve for \( dx \) in terms of \( du \). Thus, \( dx = \frac{du}{3x^2} \).

Step 4: Substitute in the Integral

Substitute \( u \) and \( dx \) in terms of \( du \) into the integral: \( \int \frac{x^2}{(x^3 - 3)^2} \, dx \) becomes \( \int \frac{x^2}{u^2} \cdot \frac{du}{3x^2} \).

Step 5: Simplify the Integrand

The \( x^2 \) terms cancel out in the integrand, simplifying the expression to \( \frac{1}{3} \int u^{-2} \, du \).

Step 6: Integrate with Respect to u

Integrate the simplified form: \( \frac{1}{3} \int u^{-2} \, du = \frac{1}{3} \left(-u^{-1}\right) + C = -\frac{1}{3u} + C \).

Step 7: Substitute Back in Terms of x

Replace \( u \) with \( x^3 - 3 \) using the initial substitution: \( -\frac{1}{3(x^3 - 3)} + C \).

Key Concepts

Variable Substitution

Variable substitution is a key strategy in integration. It is like changing the language of the function to make it better to work with. For indefinite integrals, this technique helps transform a complicated integral into a simpler one. Here, you identify a part of the integrand to replace it with a single variable, usually denoted by \( u \). This new variable 'substitutes' parts of the integral, offering a fresh perspective.

To apply variable substitution effectively, follow these steps:

• Identify a part of the integrand that can be replaced with \( u \). Often, this is an expression whose derivative is also part of the integrand.
• Differentiate \( u \) with respect to \( x \) to find \( du \).
• Rearrange the expression \( du = \text{{expression}} \, dx \), solving for \( dx \).

This method can significantly simplify your integral, as shown in the given exercise with \( u = x^3 - 3 \). By substituting \( u \) into the integral, we eliminate complex expressions, making it more accessible.

Integration Techniques

Integration techniques are methods used to solve integrals that are not straightforward. Some common techniques include substitution, integration by parts, and trigonometric integration.

In the context of this exercise, substitution was the chosen technique. After identifying \( u = x^3 - 3 \), the problem required the conversion of all \( x \)-related terms to \( u \). This involves:

• Cancelling like terms from the numerator and denominator where possible.
• Transforming the differential \( dx \) properly using \( du \) as derived.

Finally, integrating the new simpler expression with respect to \( u \), gives us a result in terms of \( u \). This particular technique relies on reducing the integral to a basic form, like \( \int u^{-2} \, du \). After integration, it is crucial to revert back to the original variable to express the solution in terms of \( x \).

Change of Variables

The change of variables is sometimes referred to as substitution in integration. It is a valuable tool for converting a difficult expression into an easier one. This method is not just about changing symbols; it involves altering the entire perspective of the problem.

For instance, when dealing with expressions like \( \int \frac{x^2}{(x^3 - 3)^2} \, dx \), noticing the term \( x^3 - 3 \) allows a change of variables to simplify the process. By expressing the integral in terms of \( u \) instead of \( x \), you're effectively redesigning the problem.

Such changes lead to rhythmic patterns:

• First, define a new variable \( u = g(x) \).
• Differential transformation follows with \( du = g'(x) \, dx \).
• Finally, solve the integral in terms of \( u \), and reverse the substitution at the end.

This strategy is instrumental for tackling integrals that initially appear intimidating, simplifying their structure and easing the computation.