Problem 5 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int x \sqrt{2 x-3} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{10}(2x-3)^{5/2} + \frac{1}{2}(2x-3)^{3/2} + C\).

Step 1: Identify the Integral Form

Look through the table of integrals at the back of the book to find a form that matches or is similar to \(\int x \sqrt{2x-3}\, dx\). This integral could potentially fit a form that involves substitution.

Step 2: Choose a Substitution

Observe that the expression under the square root, \(2x-3\), suggests using a substitution. Let \(u = 2x - 3\). This choice simplifies the square root part of the integral.

Step 3: Differentiate the Substitution

Differentiate \(u = 2x - 3\) with respect to \(x\). This gives \(\frac{du}{dx} = 2\), which implies \(dx = \frac{1}{2}du\).

Step 4: Express x in terms of u

Since \(u = 2x - 3\), solve for \(x\) in terms of \(u\): \(x = \frac{u+3}{2}\). Substitute this expression into the integral.

Step 5: Substitute into the Integral

Substitute \(x = \frac{u+3}{2}\) and \(dx = \frac{1}{2}du\) into the integral. The integral becomes: \(\int \frac{u+3}{2} \sqrt{u} \cdot \frac{1}{2} \, du = \frac{1}{4} \int (u + 3) u^{1/2} \, du\).

Step 6: Simplify the Integral

Simplify \(\frac{1}{4} \int (u + 3) u^{1/2} \, du\). Distribute \(u^{1/2}\) to get: \(\frac{1}{4} \int (u^{3/2} + 3u^{1/2}) \, du\).

Step 7: Solve the Simplified Integral

Solve each part separately: \(\int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5}u^{5/2}\) and \(\int 3u^{1/2} \, du = 3 \frac{u^{3/2}}{3/2} = 2u^{3/2}\).

Step 8: Combine Results and Simplify

Combine the results and factor out the constant: \(\frac{1}{4} \left( \frac{2}{5}u^{5/2} + 2u^{3/2} \right) = \frac{1}{10}u^{5/2} + \frac{1}{2}u^{3/2}\).

Step 9: Substitute Back

Substitute back \(u = 2x - 3\) into the expression: \(\frac{1}{10}(2x - 3)^{5/2} + \frac{1}{2}(2x - 3)^{3/2}\).

Step 10: Add the Constant of Integration

Finally, add the constant of integration \(C\) to the result, giving: \(\frac{1}{10}(2x - 3)^{5/2} + \frac{1}{2}(2x - 3)^{3/2} + C\).

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique in calculus, especially useful for simplifying integrals involving complex expressions. When confronted with an integral that seems difficult to solve directly, substitution can often transform it into a more manageable form.

To perform integration by substitution, follow these general steps:

• Identify a part of the integrand that can be substituted with a simpler variable, usually denoted as \(u\).
• Express \(x\) and \(dx\) in terms of \(u\) and \(du\). This often involves basic algebraic manipulations and differentiation.
• Rewrite the integral in terms of \(u\), which should ideally simplify the expression.
• Perform the integration with respect to \(u\). Once integrated, substitute back the original variable \(x\) to obtain the final result.

This method is particularly effective when dealing with functions under radicals or exponentials, where substitution can remove the direct complexity of the original function.

In the given problem, the substitution \(u = 2x - 3\) allowed us to express the square root function in terms of \(u\), simplifying the integrand and facilitating integration.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial in integral calculus. Both concepts form the backbone of calculus applications, from computing areas to solving differential equations.

Definite integrals are used to calculate the area under a curve within a specific range. It has both upper and lower limits and provides a numeric value reflecting the total area. The notation for a definite integral is generally expressed as \(\int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the limits of integration. After integration, evaluate the antiderivative at these limits and subtract the results.

Indefinite integrals, however, represent families of functions and are expressed without specified limits. They have the form \(\int f(x) \, dx\), resulting in an antiderivative plus a constant of integration \(C\). This constant represents the infinite number of vertical translations possible for the antiderivative function.

In our example, we're dealing with an indefinite integral since no limits are provided. The step-by-step method, closing with adding \(C\), ensures all solutions represent the general form of the antiderivative.

Table of Integrals

A table of integrals is a valuable resource for solving complex integrals quickly. It contains a list of commonly encountered integrals and their solutions, saving time and effort to perform integration repeatedly by hand.

When tackling an integration problem, refer to the table to find a matching form of the integral you are trying to solve. Integration by substitution often uses this approach, substituting into a simplified form found in the table, before solving the integral.

In our case, even though we performed manual substitution and integration, initially consulting a table can guide us to an appropriate substitution or confirm the solution. Having the ability to cross-reference enhances confidence in solving integrals and speeds up the process.

By familiarizing oneself with these tables, you can often spot potential substitutions or direct integrals immediately, reinforcing the integration tools at your disposal.