Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the method of substitution to find each of the following indefinite integrals. $$ \int x\left(x^{2}+3\right)^{-12 / 7} d x $$

Step-by-Step Solution

Answer

The integral is \(-\frac{7}{10} (x^2 + 3)^{-5/7} + C\).

Step 1: Identify the Inner Function

The integration problem suggests substitution. Examine the expression inside the parentheses: \( x^2 + 3 \). This is where we apply the substitution technique, setting \( u = x^2 + 3 \). This new variable \( u \) simplifies the integration process.

Step 2: Differentiate to Find du

Differentiate the chosen substitution \( u = x^2 + 3 \) with respect to \( x \). This gives: \( \frac{du}{dx} = 2x \). Rearrange this equation to find \( du \) in terms of \( dx \): \( du = 2x \, dx \). This helps relate the differentials in terms of \( u \).

Step 3: Solve for dx in Terms of u

From \( du = 2x \, dx \), solve for \( dx \): \( dx = \frac{du}{2x} \). This substitution will replace \( dx \) in the integral.

Step 4: Substitute and Simplify the Integral

Replace \( x^2 + 3 \) with \( u \) and \( dx \) with \( \frac{du}{2x} \) in the integral: \[ \int x \left(u^{-12/7}\right) \left(\frac{du}{2x}\right) \]. Simplify the expression by canceling \( x \) in the numerator and denominator, leading to: \( \frac{1}{2} \int u^{-12/7} du \).

Step 5: Integrate with Respect to u

Integrate the simplified integral: \( \frac{1}{2} \int u^{-12/7} du \). The antiderivative of \( u^n \) is \( \frac{u^{n+1}}{n+1} \), so apply it here: \[ \frac{1}{2} \cdot \frac{u^{-12/7 + 1}}{-12/7 + 1} + C \]. Simplify the exponent \(-12/7 + 1 = -5/7\) and calculate the integral: \(-\frac{7}{2 \cdot 5} u^{-5/7} + C\).

Step 6: Substitute Back for x

Replace \( u \) with the original expression \( x^2 + 3 \): \[-\frac{7}{10} (x^2 + 3)^{-5/7} + C\]. This is the indefinite integral in terms of \( x \).

Key Concepts

Indefinite Integrals

Indefinite integrals are an essential concept in calculus used to find a general form of an antiderivative for a given function. They are expressed without specified limits of integration, meaning they represent a family of functions. The notation for an indefinite integral is similar to a summation with the symbol \( \int \), followed by the function to integrate and the differential, typically denoted by \( dx \). For example, \( \int f(x) \, dx \) indicates the integration of \( f(x) \) with respect to \( x \).

Indefinite integrals include a constant of integration, often symbolized by \( C \), because the derivative of a constant is zero. This constant accounts for the fact that there are infinitely many antiderivatives, and it reflects vertical shifts in the graph of the antiderivative.

• The goal is to retrieve a function whose derivative yields the integrand.
• Calculated without particular bounds, leading to a general solution.
• Always includes a constant of integration.

Integration Techniques

Integration techniques are approaches used to solve integrals that might not be straightforward. One of the most common techniques is substitution. This method is especially useful for integrals containing composite functions. The idea is to simplify the integral by making a substitution that transforms the complex expression into a recognizable and easier-to-integrate form.

With substitution, the following steps are usually applied:

• Identify a portion of the integrand to replace with a substitution variable, typically \( u \).
• Differentiate the substitution variable to relate it back to the original variable.
• Rewrite the integral in terms of the new variable \( u \), simplifying it if possible.
• Integrate with respect to \( u \).
• Finally, replace \( u \) with the initial expression to return to the original variable.

In the given exercise, the substitution \( u = x^2 + 3 \) made the integral easier by turning the focus from complex powers of a binomial to a simple power of \( u \). This technique is invaluable in calculus, reducing difficulty and improving efficiency.

Calculus Problem-Solving

Calculus problem-solving involves various strategies to manipulate and solve mathematical problems. One useful skill is recognizing when to use certain integration techniques, like substitution, which can greatly simplify the integral at hand.

Successful problem-solving in calculus requires:

• Analyzing the structure of the function: Look for patterns that suggest specific techniques.
• Performing algebraic manipulation: Rewriting expressions can make complex integrals manageable.
• Checking results: Use differentiation to verify the correctness of your antiderivative.
• Practicing regularly: Continual practice helps build intuition for choosing the best technique efficiently.

In this exercise, recognizing the structure of \( x(x^2 + 3)^{-12/7} \) allowed the successful use of substitution. Identifying \( x^2 + 3 \) as a single unit facilitated the simplification and solution of the integral, reflecting a typical calculus problem-solving approach.