Problem 46 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} \frac{2 x}{\left(4 x^{2}+2\right)^{1 / 3}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to approximately 1.654875.

Step 1: Choose a Substitution

First, we will choose an appropriate substitution to simplify the integral. Since the denominator contains the expression \( (4x^2 + 2)^{1/3} \), we can set \( u = 4x^2 + 2 \). This substitution will simplify the expression.

Step 2: Calculate the Derivative

Compute the derivative of \( u = 4x^2 + 2 \) with respect to \( x \). This yields \( \frac{du}{dx} = 8x \). Therefore, \( du = 8x \, dx \).

Step 3: Express dx in terms of du

We need to express \( dx \) in terms of \( du \) and \( x \). From \( du = 8x \, dx \), we have \( dx = \frac{du}{8x} \).

Step 4: Rewrite the Integral with the Substitution

Substitute \( u = 4x^2 + 2 \) and the expression for \( dx \) into the integral:\[\int \frac{2x}{(4x^2 + 2)^{1/3}} \, dx = \int \frac{2x}{u^{1/3}} \cdot \frac{du}{8x} = \frac{1}{4} \int u^{-1/3} \, du\]

Step 5: Evaluate the New Integral

Evaluate the integral \( \int u^{-1/3} \, du \). The antiderivative is \( \frac{u^{2/3}}{2/3} = \frac{3}{2} u^{2/3} \). So, the integral becomes:\[\frac{1}{4} \cdot \frac{3}{2} u^{2/3} = \frac{3}{8} u^{2/3} \]

Step 6: Change Limits of Integration

Calculate the new limits of integration using the substitution. When \( x = 0 \), \( u = 4(0)^2 + 2 = 2 \), and when \( x = 2 \), \( u = 4(2)^2 + 2 = 18 \). Thus, the limits of integration change from \( 0 \) and \( 2 \) to \( 2 \) and \( 18 \).

Step 7: Evaluate the Definite Integral

Substitute \( u = 18 \) and \( u = 2 \) into \( \frac{3}{8} u^{2/3} \) and compute:\[\frac{3}{8} [18^{2/3} - 2^{2/3}]\]Calculate separately: \( 18^{2/3} \approx 6 \) and \( 2^{2/3} \approx 1.587 \). Subtract and multiply by \( \frac{3}{8} \) to obtain:\[\frac{3}{8} (6 - 1.587) \approx \frac{3}{8} \times 4.413 = 1.654875\]

Step 8: Final Answer

The definite integral of \( \frac{2x}{(4x^2 + 2)^{1/3}} \) from \( 0 \) to \( 2 \) with the substitution method is approximately equal to 1.654875.

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus used to evaluate integrals by simplifying complex expressions. This approach transforms the original variable, often denoted as \(x\), into a new variable, \(u\). This makes the integral easier to solve. Let's break down how it works and why it's useful.

• The main goal is to rewrite the integral in terms of \(u\). This simplifies the equation by removing intricate functions and making calculations straightforward.
• A suitable substitution is crucial. It often involves setting \(u\) equal to a function inside the integral, such as an expression in the denominator or inside a radical.

In the given exercise, the substitution \(u = 4x^2 + 2\) was chosen because it simplified the denominator \((4x^2 + 2)^{1/3}\). This led to transforming both the bounds of integration and the differential \(dx\), allowing the integral to be computed as \( \int u^{-1/3} \, du\). Converting back to the original variable is typically the final step after evaluating the integral.

Calculus

Calculus is a branch of mathematics focused on rates of change (differentiation) and accumulation of quantities (integration). Its applications span multiple fields, such as physics, engineering, and economics. Let's understand how calculus is applied when solving integrals:

• Differentiation involves understanding how functions change. It's foundational in finding derivatives, which are crucial in the substitution method for expressing \(dx\) in terms of \(du\).
• Integration is essentially the reverse process of differentiation. It measures the accumulation of areas under curves or other cumulative quantities.

In this exercise, calculus principles help transform the complicated original integral into a simpler form through differentiation (finding \(du\)) and integration. Solving the definite integral provides the total area under the curve \(\frac{2x}{(4x^2 + 2)^{1/3}}\) from 0 to 2.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function we were integrating. Finding antiderivatives is a fundamental task in calculus for computing definite integrals. Here’s what to keep in mind:

• An antiderivative of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). The \(dx\) indicates integration with respect to \(x\).
• Once you find an antiderivative, it can be used to calculate definite integrals over a range by evaluating the antiderivative at the upper and lower bounds of the interval.

In the given problem, the antiderivative \(\frac{3}{2}u^{2/3}\) was determined using standard integration rules. This helps compute the definite integral by substituting back the original variable and the limits of integration \(x = 2 \) and \(x = 0\), allowing for the approximate area under the curve from 0 to 2 to be calculated.