Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{3} \frac{x^{2}+1}{\sqrt{x^{3}+3 x}} d x $$

Short answer

The value of the definite integral is \(\frac{8}{3}\).

Step-by-step solution

Choose a Substitution

We need to simplify the integrand by choosing an appropriate substitution. Notice that the denominator involves the expression \(x^3 + 3x\). Let's try a substitution where \(u = x^3 + 3x\).

Find the Derivative of the Substitution

Differentiate \(u = x^3 + 3x\) with respect to \(x\) to find \(du\). We get \(du = (3x^2 + 3) \, dx = 3(x^2 + 1) \, dx\). Let's solve for \(dx\).

Solve for dx

From \(du = 3(x^2 + 1) \, dx\), we have \(dx = \frac{du}{3(x^2 + 1)}\). This allows us to substitute \(dx\) in the integral.

Substitute in the Integral

Substitute \(u\) and \(dx\) back into the integral:\[\int_{1}^{3} \frac{x^2+1}{\sqrt{x^3+3x}} \, dx = \int \frac{1}{\sqrt{u}} \, \frac{du}{3}\].

Simplify and Change the Limits of Integration

Simplify the integral using the coefficient to find:\[\int \frac{1}{3\sqrt{u}} \, du\].Now, change the limits of integration: when \(x = 1\), \(u = 1^3 + 3(1) = 4\); when \(x = 3\), \(u = 3^3 + 3(3) = 36\). Thus, our new limits are from 4 to 36.

Integrate

Now integrate with respect to \(u\):\[\int_{4}^{36} \frac{1}{3\sqrt{u}} \, du = \frac{1}{3} \times 2\sqrt{u} \bigg|_{4}^{36}\].

Evaluate the Definite Integral

Evaluate the integral with the new limits:\[\frac{1}{3}(2\sqrt{36} - 2\sqrt{4}) = \frac{1}{3}(12 - 4) \].Calculate the final result:

Calculate the Final Value

\[\frac{1}{3} \times 8 = \frac{8}{3}\].Thus, the value of the definite integral is \(\frac{8}{3}\).

Key concepts

Integration Techniques

Integration techniques are fundamental tools in calculus and are employed to solve integrals that cannot be evaluated by basic methods. These techniques provide systematic approaches to simplify and compute integrals, especially complex ones. Commonly used methods include:

• Substitution
• Integration by parts
• Partial fraction decomposition
• Trigonometric substitution

In the context of our exercise, we use the substitution method. This method involves transforming a challenging integral into a simpler form by substituting part of the integrand with a new variable. This often makes the integral more manageable and easier to solve.

Definite Integrals

Definite integrals are used to compute the net area under a curve between two specified points. They provide the exact accumulation of quantities, which can represent work done by a force, area, volume, etc.
When evaluating definite integrals, you need to consider the limits of integration. These limits are the values between which you want to find the area.
Definite integrals are calculated by finding the antiderivative and then applying the Fundamental Theorem of Calculus, which links the concept of differentiation with integration. After substituting back any replaced variable, we evaluate the antiderivative at the upper limit and subtract its evaluation at the lower limit, to find the final result.

Mathematical Substitution

Mathematical substitution is a pivotal technique in solving complex integrals. It involves replacing a section of the integral with a simpler expression. This is achieved by defining a new variable, say, \( u \), and expressing part of the integrand in terms of \( u \).
In our example, we chose \( u = x^3 + 3x \). This choice simplifies the expression, making \( \int \frac{1}{\sqrt{u}} \, du \) more straightforward to solve.

The Process of Substitution

• Identify a part of the integral that can be substituted.
• Express the chosen part in terms of a new variable (\( u \)).
• Find the derivative (\( du \)) of this new expression.
• Solve for \( dx \) and replace in the integral.
• Don’t forget to change the limits of integration if dealing with definite integrals.

Calculus Problem Solving

Calculus problem solving often involves integrating various mathematical methods to arrive at a solution. It's key to understand which techniques can be combined to simplify the process.
In our exercise, we emphasized using substitution, which turned a complex-form integral into a simpler one. Here's how this approach is typically structured:

Steps in Problem Solving

• Identify suitable techniques: Assessment of the problem to decide the best integration strategy.
• Implementing substitution: Applying the technique to introduce a new variable to simplify the integrand.
• Changing limits: In definite integrals, ensuring that when you substitute, the limits of integration reflect the new variable's context.
• Integrating: Solving the modified integral and simplifying the expression.
• Substituting back: For indefinite integrals, reintroducing the original variable; evaluating definite integrals with the updated limits.

By following these structured steps, solving calculus problems becomes more accessible and less error-prone.