Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the method of substitution to find each of the following indefinite integrals. $$ \int x^{2}\left(x^{3}+5\right)^{9} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{(x^3 + 5)^{10}}{30} + C \).

Step 1: Identify the Substitution

To use the substitution method, we start by selecting a substitution that simplifies the integral. Here, let us choose the inner function of the composite expression as the substitution. Set \( u = x^3 + 5 \).

Step 2: Compute the Derivative of the Substitution

Differentiate the substitution equation with respect to \( x \) to link it back to the original integral. Calculating the derivative, we have \( \frac{du}{dx} = 3x^2 \), which implies \( du = 3x^2 \, dx \).

Step 3: Solve for dx

We need to express \( dx \) in terms of \( du \) and \( x \). Rearranging \( du = 3x^2 \, dx \), we get \( dx = \frac{du}{3x^2} \).

Step 4: Substitute in the Integral

Replace all instances of \( x \) and \( dx \) in the integral with \( u \) and \( du \) respectively. The integral becomes \( \int x^2 (u)^9 \frac{du}{3x^2} \), which simplifies to \( \int \frac{u^9}{3} \, du \).

Step 5: Integrate with Respect to u

Integrate \( \frac{1}{3} \, u^9 \) with respect to \( u \). The integral becomes \( \frac{1}{3} \cdot \frac{u^{10}}{10} + C \), or equivalently \( \frac{u^{10}}{30} + C \).

Step 6: Back Substitute to x

Replace \( u \) back with \( x^3 + 5 \) to express the result in terms of the original variable. The final integral becomes \( \frac{(x^3 + 5)^{10}}{30} + C \).

Key Concepts

Indefinite Integral

An indefinite integral, often symbolized by the integral sign \( \int \), represents a family of functions that includes all the antiderivatives of a given function. Unlike definite integrals, indefinite integrals do not have upper and lower limits. You might often see them written with a constant \( C \), which indicates the presence of an arbitrary constant in the family of solutions.

• Symbol: \( \int f(x) \, dx \)
• No upper or lower limits
• Includes a + \( C \) to indicate all possible constants

The main goal when computing an indefinite integral is to find the original function whose derivative matches the integrand (the function inside the integral sign). In our exercise, the goal is to find the antiderivative of \( x^2(x^3+5)^9 \), and we accomplish this using the substitution method.

Composite Function

A composite function is created when one function is applied inside another. With a composite, you have an inner function and an outer function. In the exercise, \( x^3 + 5 \) is the inner function and it is "wrapped" by the outer function \( (x^3 + 5)^9 \).

• Inner function: \( x^3 + 5 \)
• Outer function: \( (x^3 + 5)^9 \)

Composite functions are central in the substitution method, where you select the inner function as the substitution variable \( u \). This makes the integration process much easier and more efficient, especially when the inner function is more complicated and its derivative is part of the outer function.

U-Substitution

U-substitution is a technique used to simplify finding the integrals of composite functions. The idea is to replace a complicated part of the integral with a simpler variable, typically \( u \). Here, you identify a section of the integrand that can be substituted with \( u \).In our exercise:

• We let \( u = x^3 + 5 \), simplifying \( (x^3+5)^9 \) as \( u^9 \).
• Then, differentiate \( u \) to find \( du \) in terms of \( dx \): \( du = 3x^2 \, dx \).
• Rearrange it to find \( dx \) in terms of \( du \), giving \( dx = \frac{du}{3x^2} \).

By substituting \( x^3 + 5 \) with \( u \), and \( dx \) with \( \frac{du}{3x^2} \), the integral becomes easier to handle. This results in integrating \( \frac{u^9}{3} \) with respect to \( u \).

Derivative Computation

In calculus, finding the derivative of a function means computing how the function changes at any point along its curve. Derivative computation is critical in substitution because it allows you to replace \( dx \) in the integral with something involving \( du \).In the context of this exercise, differentiation plays a key role:

• Once \( u = x^3 + 5 \) is determined, differentiate \( u \) by finding \( \frac{du}{dx} \).
• This gives us \( \frac{du}{dx} = 3x^2 \), simplifying to \( du = 3x^2 \, dx \).

The differentiation step is crucial because it provides the link back to the original variable \( x \), allowing the entire expression \( dx \) to be expressed in terms of \( du \). It ensures the conversion of the entire integral into a form that is both workable and solvable. After performing the integral calculation, the process is completed by back substituting \( u \) to convert it back to terms of \( x \).