Question

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x $$

Short answer

The result of the definite integral is \( \frac{1}{3}(22^{1/2} - (-5)^{1/2})\).

Step-by-step solution

Identify the suitable substitution

Look for a function and its derivative within the integral. By observation \(u=x^{2}-8\) is a promising substitution because it's derivative \(2x \(, matches the \(x\) in the numerator.

Perform the substitution

With \(u=x^{2}-8\), then \(du = 2x \,dx\). Adjusting for the extra factor of 2, \(dx = du/(2x)\). Replace \(x\) in terms of \(u\) and \(dx\) in the integral, we get \(\int_{3}^{6} \frac{x}{3 \sqrt{u}} * \frac{du}{2x}\) which simplifies to \(\int_{3}^{6} \frac{du}{6 \sqrt{u}}\).

Evaluate the new integral

The new integral is a simpler standard form. Evaluate it using reverse power rule:\(\frac{1}{6}\) \(\int_{3}^{6} \) \(u^{-1/2} du = \frac{1}{6} * 2 * u^{1/2}\)

Substitute `u` back and Use the Fundamental Theorem of Calculus

Substitute \(u\) back into the result as \(x^{2}-8\), and recall the limits are on \(x\), not \(u\). \(\frac{1}{3}*(x^{2}-8)^{1/2}\Bigg]_{3}^{6}\) = \(\frac{1}{3}(22^{1/2} - (-5)^{1/2})\)

Key concepts

Symbolic Integration

Symbolic integration involves finding the integral of a function in its analytic form, rather than numerically evaluating it. In this exercise, we aim to calculate the definite integral of a given function using symbolic reasoning and algebraic manipulation.
When performing symbolic integration:

• We look for patterns and standard forms in the integrand that suggest an algebraic technique.
• We may use known antiderivatives of functions to solve the problem.

Symbolic integration can be aided by software tools, but understanding the theory provides valuable intuition. For instance, recognizing when to apply a substitution can simplify an otherwise complex integral. This exercise demonstrates how symbolic integration can turn an intimidating definite integral into something manageable by breaking it down into simpler, standard parts.

Substitution Method

The substitution method is a technique in calculus used to simplify and solve integrals. It involves substituting part of the integrand with a new variable, making the integral easier to evaluate. In our exercise, notice how we chose the substitution to simplify the integral significantly. Here’s how it works:

• First, identify a substitution where the derivative of your new variable is present within the integral. For example, we chose the substitution \( u = x^2 - 8 \).
• Next, calculate the differential, \( du = 2x \,dx \), and rearrange to find \( dx \) in terms of \( du \) and \( x \).

This step transforms the integral into a more straightforward form. The substitution method is crucial because it can turn complex functions into simpler ones, drastically reducing the complexity of evaluation. By substituting wisely, we change the integral from one difficult to solve to one that uses standard calculus techniques.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, two core operations in calculus. It states that if a function is continuous over an interval, then the integral of its derivative over that interval returns the net change of the function over that interval.
This theorem is invaluable for evaluating definite integrals after finding antiderivatives. Once we simplify our integral using substitution, the next step is evaluating it over the specified limits by using the antiderivative. Remember:

• Find the antiderivative of the transformed expression.
• Substitute back any variables used in substitution to return to the original variable.
• Apply the limits of integration to find the net change.

The exercise demonstrates the Fundamental Theorem’s strength: once we've evaluated the antiderivative and returned to the original variables, we simply apply the limits of the integral to find the solution, making it a practical and powerful method for solving real-world problems.