Problem 16 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int \sqrt[3]{2 x-4} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{3}{8} (2x - 4)^{4/3} + C \).

Step 1: Choose a Substitution Variable

Let's start by identifying a substitution that simplifies the integral. We notice the expression \( \sqrt[3]{2x - 4} \). A good choice of substitution would be to let \( u = 2x - 4 \).

Step 2: Differentiate the Substitution Variable

Differentiate both sides of the equation \( u = 2x - 4 \) with respect to \( x \). This gives us \( \frac{du}{dx} = 2 \), or equivalently \( du = 2 \, dx \).

Step 3: Solve for dx in Terms of du

Now solve for \( dx \) from the equation \( du = 2 \, dx \). We find \( dx = \frac{1}{2} du \).

Step 4: Rewrite the Integral in Terms of u

Substitute \( u = 2x - 4 \) and \( dx = \frac{1}{2} du \) into the original integral: \[ \int \sqrt[3]{2x - 4} \, dx = \int \sqrt[3]{u} \cdot \frac{1}{2} \, du \]This simplifies to:\[ \frac{1}{2} \int u^{1/3} \, du \]

Step 5: Integrate with Respect to u

Now integrate \( u^{1/3} \) with respect to \( u \):\[ \frac{1}{2} \int u^{1/3} \, du = \frac{1}{2} \times \frac{u^{4/3}}{4/3} + C \]Simplify to:\[ \frac{3}{8} u^{4/3} + C \]

Step 6: Substitute Back the Original Variable x

Replace \( u \) with the original expression \( 2x - 4 \):\[ \frac{3}{8} (2x - 4)^{4/3} + C \]This is the indefinite integral of the given function with respect to \( x \).

Key Concepts

Understanding the Substitution Method

The substitution method is a technique used for solving indefinite integrals that simplifies integration by changing variables. When facing a complex integral, such as the one involving a cubic root, substitution can transform it into a more manageable problem. Here's what it involves:

• Identifying a part of the integral to substitute with a single variable, often denoted as \( u \). This part usually forms a complex expression, like a radical, in the integral.
• Choosing \( u \) is strategic as it makes the integration process smoother. In our example, picking \( u = 2x - 4 \) simplifies the cubic root \( \sqrt[3]{2x - 4} \).
• Once substitution is chosen, differentiate the new variable \( u \) with respect to the original variable \( x \). For \( u = 2x - 4 \), this results in \( du = 2 \, dx \).
• Express \( dx \) in terms of \( du \), leading to \( dx = \frac{1}{2} \, du \), which is key for integrating in terms of \( u \).
• Replace occurrences of the original variable with the substitute, reformulate the integral, and proceed with integration in terms of \( u \). This makes the original complex integral easier to work with.

The substitution method is powerful, particularly for integrals involving radicals, products, or quotients, often resulting in an elementary integral that is straightforward to integrate.

Exploring Cubic Roots in Integration

Cubic roots, like square roots, present a unique challenge in integration due to their non-linear nature. Let's take a look at what happens when you encounter a cubic root in an integral:

• The integral \( \int \sqrt[3]{2x - 4} \, dx \) presents a cubic root that complicates direct integration due to its fractional exponent form, \( \int (2x - 4)^{1/3} \, dx \).
• Cubic roots have the form \( x^{1/3} \), representing the third root of \( x \). For integration purposes, fractional exponents are useful, transforming the root into a power function that can be integrated using basic power rule.
• To integrate a power function, use \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), assuming \( n eq -1 \). This rule applies to any real number exponent, including fractions found in roots.
• By transforming the integral into a function of \( u \) as shown in the substitution method, the cubic root \( \sqrt[3]{u} \) becomes \( u^{1/3} \).

Incorporating substitution with cubic roots often results in an integral that's ready for the power rule. Understanding this transformation is key to managing integrals involving roots.

Integration Steps in Solving Indefinite Integrals

Solving indefinite integrals through substitution involves a series of clear steps that make the process systematic. Let's break down these steps using the example from the original exercise:**1. Choose a Substitution Variable**: Start by identifying an appropriate substitution. For the integral \( \int \sqrt[3]{2x - 4} \, dx \), pick \( u = 2x - 4 \) to simplify the expression.**2. Compute the Derivative**: Differentiate \( u \) with respect to \( x \), yielding \( du = 2 \, dx \). This calculation will help express the remaining parts of the integral in terms of \( u \).**3. Solve for \( dx \) in terms of \( du \)**: Rearrange the equation to get \( dx = \frac{1}{2} \, du \). Substituting \( dx \) is essential for reformulating the integral correctly.**4. Rewrite the Integral**: Substitute the \( u \) expression and derived \( dx \) back into the integral. Here, \( \int \sqrt[3]{2x - 4} \, dx \) turns into \( \frac{1}{2} \int u^{1/3} \, du \).**5. Integrate with Respect to \( u \)**: Apply the power rule. Integrate \( \frac{1}{2} \int u^{1/3} \, du \) to obtain \( \frac{3}{8} u^{4/3} + C \).**6. Substitute Back the Original Variable**: Replace \( u \) with \( 2x - 4 \) as it was initially the substitution variable. The solution is \[ \frac{3}{8} (2x - 4)^{4/3} + C \].Following these steps ensures a structured approach, making even complex integrals manageable. Mastering this method unlocks a wide array of integral problems.