Problem 43 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral evaluates to approximately 10.2076.

Step 1: Choose a Substitution

Select a substitution to simplify the integral. Let\( u = x^2 + 1 \). This implies that\( du = 2x \, dx \) must hold.

Step 2: Express dx in terms of du

Solve for\( dx \) by rearranging\( du = 2x \, dx \). Obtain\( dx = \frac{du}{2x} \).

Step 3: Rewrite the Integral

Substitute\( u = x^2 + 1 \) and\( dx = \frac{du}{2x} \) into the integral. The expression becomes\[ \int x \sqrt{u} \cdot \frac{du}{2x}. \] Simplify to\[ \frac{1}{2} \int \sqrt{u} \, du. \]

Step 4: Change the Limits of Integration

Change the limits to reflect the substitution. When\( x = 0, u = 0^2 + 1 = 1 \) and when\( x = 3, u = 3^2 + 1 = 10 \). The modified limits are from 1 to 10.

Step 5: Evaluate the Integral

Integrate\( \frac{1}{2} \int_1^{10} u^{1/2} \, du \). The antiderivative of\( u^{1/2} \) is\( \frac{2}{3}u^{3/2} \). Evaluate from 1 to 10:\[ \frac{1}{2} \left[ \frac{2}{3}u^{3/2} \right]_1^{10} = \frac{1}{2} \cdot \frac{2}{3} \left(10^{3/2} - 1^{3/2}\right). \]

Step 6: Compute the Final Answer

Calculate\( 10^{3/2} = \sqrt{10^3} = 31.6228 \) and\( 1^{3/2} = 1 \). So the expression inside gives\( 31.6228 - 1 = 30.6228 \). Thus, \[ \frac{1}{2} \cdot \frac{2}{3} \cdot 30.6228 = \frac{1}{3} \cdot 30.6228 \approx 10.2076. \]

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus for simplifying integrals. By changing variables, we make it easier to integrate certain functions.
When tackling an integral, we can choose a new variable that simplifies the integrand. Here's how:

• Pick a substitution: typically, identify part of the integrand that, when substituted, will result in a simpler form.
• In this exercise, we chose \( u = x^2 + 1 \), making the square root expression easier to handle.
• Find the differential \( du \) in terms of the original variable and express \( dx \).
• Rewrite the original integral with the substitution.

This swap of variables transforms a complex integral into something more manageable.

Limits of Integration

Changing variables in an integral modification also affects the limits of integration. We adjust the limits to match our substitution. This is essential in definite integrals, where everything stays consistent.
To change limits of integration:

• Calculate the new limits using the substitution formula.
• In our exercise, substituting \( u = x^2 + 1 \) transforms the original limits \( 0 \) and \( 3 \) to \( 1 \) and \( 10 \) respectively.

By doing this, we ensure that our definite integral accurately reflects the change in variables.
It also simplifies the computation by allowing the substitution to directly influence the evaluated limits.

Antiderivative

Finding an antiderivative is an essential step in solving definite integrals. It involves computing the inverse operation of taking a derivative.
Here's how we handle it:

• Identify the expression to integrate, as impacted by the substitution.
• For \( u^{1/2} \), its antiderivative is \( \frac{2}{3}u^{3/2} \).
• Integrating effectively means finding this function that, when differentiated, returns the original integrand.
• Always apply the antiderivative within the new limits of integration that reflect your substitution.

In this manner, we transform the problem to one of basic integration, then apply those results to derive the precise value of the definite integral.

Calculus Problem Solving

Solving calculus problems involves a blend of several techniques. Each plays a vital role in different contexts. Here are the general steps:

• Understand the problem: Break down the integral and determine the best method – such as substitution, integration by parts, or partial fractions.
• Choose the right tools and use them effectively: Here, substitution and determinate integration were used to ease the calculations.
• Carry out each step carefully: Ensure the changes in the integral align with any new variable adjustments, such as limits of integration.
• Apply mathematical precision: Include adjustments, like converting the new limits correctly, and solve using the derived antiderivative.

By following a clear and systematic approach, calculus problems become much less intimidating.
This method also ensures precise and accurate results for various complex integration problems.