Problem 57 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{1} x^{2} \sqrt{x^{3}+1} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{4\sqrt{2} - 2}{9} \).

Step 1: Choose a Substitution

To solve the integral \( \int_{0}^{1} x^{2} \sqrt{x^{3}+1} \, dx \), we start by choosing a suitable substitution. Let \( u = x^3 + 1 \). This implies that \( du = 3x^2 \, dx \), or rearranging gives \( x^2 \, dx = \frac{1}{3} \, du \).

Step 2: Change Limits of Integration

Now we change the limits of integration from \( x \) to \( u \). When \( x = 0 \), \( u = 0^3 + 1 = 1 \). When \( x = 1 \), \( u = 1^3 + 1 = 2 \). Thus, the new limits for \( u \) are from 1 to 2.

Step 3: Substitute into the Integral

Substitute \( u \, \text{and} \, du \) into the integral: \[ \int_{0}^{1} x^{2} \sqrt{x^{3}+1} \, dx = \int_{1}^{2} \sqrt{u} \cdot \frac{1}{3} \, du. \] This simplifies to \( \frac{1}{3} \int_{1}^{2} u^{1/2} \, du \).

Step 4: Integrate with Respect to \( u \)

Now, integrate \( \frac{1}{3} \int_{1}^{2} u^{1/2} \, du \). Using the power rule for integration, the integral of \( u^{1/2} \) is \( \frac{2}{3}u^{3/2} \). Thus, \[ \frac{1}{3} \cdot \left[ \frac{2}{3}u^{3/2} \right]_{1}^{2} = \frac{2}{9} \left[ u^{3/2} \right]_{1}^{2}. \]

Step 5: Evaluate the Definite Integral

Substitute the limits of integration into the expression: \[ \frac{2}{9} \left[ (2^{3/2}) - (1^{3/2}) \right] = \frac{2}{9} [ \sqrt{8} - 1 ]. \] Since \( \sqrt{8} = 2\sqrt{2} \), the expression becomes \[ \frac{2}{9} (2\sqrt{2} - 1). \]

Step 6: Simplify the Expression

Simplify \( \frac{2}{9} (2\sqrt{2} - 1) \) to give the final result. The expression simplifies to: \( \frac{4\sqrt{2} - 2}{9}. \)

Key Concepts

Substitution Method

The substitution method is a technique used in calculus to simplify the process of finding an integral. It is particularly useful when dealing with complex expressions that are difficult to integrate directly. The basic idea is to introduce a substitution that transforms the original integral into a simpler form. For example, in the given problem, we choose a substitution that simplifies the expression under the integral sign.To apply the substitution method:

• Select a substitution that will simplify part of the integrand. This usually involves substituting a complex part of the expression with a single variable.
• Define this new variable, say, let \( u = x^3 + 1 \). This substitution simplifies the square root found in the original problem.
• Differentiate the expression for \( u \) with respect to \( x \) to find \( du \) in terms of \( dx \). In this example, \( du = 3x^2 \, dx \).
• Rearrange to express \( x^2 \, dx \) in terms of \( du \), giving us \( x^2 \, dx = \frac{1}{3} du \).

By making these substitutions, the integration becomes much more straightforward. It allows you to focus on a simpler integral form, which is easier to compute.

Integration Limits

When applying the substitution method to definite integrals, it's crucial to also change the limits of integration. This step ensures calculations remain consistent and aligned with the substitution. Initially, limits are derived from the variable of integration, \( x \), but after substitution, they become bounds for the new variable, \( u \).Here's how to adjust the integration limits:

• Determine the new limits of integration by substituting the old limits into your substitution equation. For instance, if \( x = 0 \), then \( u = 0^3 + 1 = 1 \).
• Similarly, for the upper limit, substitute \( x = 1 \) into \( u = x^3 + 1 \), resulting in \( u = 1^3 + 1 = 2 \).

These new limits, \( u = 1 \) to \( u = 2 \), replace the original \( x \) limits in the integral expression. This ensures the integral computation reflects the correct range for the new variable.

Power Rule for Integration

The power rule for integration is a fundamental principle used to integrate functions of the form \( x^n \). This rule states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( n \) is not equal to -1.Applying the power rule involves:

• Identifying the expression in the integral that fits the \( x^n \) form. In our transformed integral, this is \( u^{1/2} \).
• Using the rule to integrate \( u^{1/2} \) as \( \frac{2}{3} u^{3/2} \).

This integration is straightforward once the expression is simplified by substitution. The power rule helps to solve complex integrals by handling each part of the power efficiently. It is important to maintain consistency with the limits of integration and apply these at the final stage to compute the definite integral.