Problem 6 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int x(7 x+5)^{3 / 2} d x\)

Step-by-Step Solution

Answer

The integral is \( \frac{2}{343} (7x + 5)^{7/2} - \frac{2}{49} (7x + 5)^{5/2} + C \).

Step 1: Identify the Type of Integral

In this exercise, we are asked to evaluate \( \int x(7x+5)^{3/2} \, dx \). Notice that this is a single-term integral involving a polynomial multiplied by another polynomial raised to a fractional power. This setup suggests using a table of integrals or substitution method known as integration by substitution (or u-substitution) may be appropriate.

Step 2: Choose an Appropriate Substitution

We choose the inner function of the composite function as our substitution. Let \( u = 7x + 5 \). Then, differentiate both sides to find \( du \): \( \frac{du}{dx} = 7 \), thus \( dx = \frac{du}{7} \). We also express \( x \) in terms of \( u \), which gives \( x = \frac{u - 5}{7} \). This substitution simplifies the integral.

Step 3: Substitute Variables

Substitute \( x = \frac{u - 5}{7} \) and \( dx = \frac{du}{7} \) into the integral which becomes: \[ \int \frac{u - 5}{7} u^{3/2} \cdot \frac{du}{7} = \frac{1}{49} \int (u - 5) u^{3/2} \, du. \] Simplify to get: \[ \frac{1}{49} \left( \int u^{5/2} \, du - 5 \int u^{3/2} \, du \right). \]

Step 4: Integrate Using Power Rule

Use the table of integrals or power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Compute each integral separately: \[ \int u^{5/2} \, du = \frac{u^{7/2}}{7/2} = \frac{2}{7} u^{7/2}, \] and \[ 5 \int u^{3/2} \, du = 5 \cdot \frac{u^{5/2}}{5/2} = 2u^{5/2}. \] Substitute back to get: \[ \frac{1}{49} \left( \frac{2}{7} u^{7/2} - 2u^{5/2} \right). \]

Step 5: Simplify the Integral and Back-substitute

Simplify the expression: \[ \frac{1}{49} \left( \frac{2}{7} (7x + 5)^{7/2} - 2 (7x + 5)^{5/2} \right). \] Then distribute \( \frac{1}{49} \): \[ \frac{2}{343} (7x + 5)^{7/2} - \frac{2}{49} (7x + 5)^{5/2}. \] This is the evaluated integral in terms of \( x \).

Step 6: Write the Final Answer

The integral \( \int x(7x+5)^{3/2} \, dx \) is evaluated as: \[ \frac{2}{343} (7x + 5)^{7/2} - \frac{2}{49} (7x + 5)^{5/2} + C, \] where \( C \) is the arbitrary constant of integration.

Key Concepts

u-substitution

One of the fundamental techniques in integration is the use of u-substitution. It's like changing the perspective of a problem to make it easier to solve. U-substitution is particularly useful when dealing with composite functions. In simple terms, it involves selecting a part of the integrand to substitute with a new variable, typically denoted as \(u\). This can simplify the integral into a basic form that is much easier to evaluate.

Steps for u-substitution:

• Identify a part of the integrand as \(u\). Typically, this is an inner function in a composition.
• Differentiate \(u\) to find \(du\), and express \(dx\) in terms of \(du\).
• Replace all instances of the original variable with \(u\) and \(du\) in the integral.
• Simplify and integrate with respect to \(u\).
• If necessary, substitute back the original variable at the end.

Using u-substitution simplifies integrals by transforming them into a form where basic integration techniques or tables of integrals can be applied.

power rule in integration

The power rule in integration is a straightforward but powerful tool used to integrate expressions of the form \(x^n\). It states that the integral of \(x^n\), where \(n\) is any real number except \(-1\), is \(\frac{x^{n+1}}{n+1} + C\). Here, \(C\) is the constant of integration.

This rule is derived from reversing the power rule in differentiation and is one of the cornerstones of calculus.

Key Points:

• If \(n = -1\), remember to use the natural logarithm, since \(\int x^{-1} \, dx = \ln|x| + C\).
• Always add \(C\) for indefinite integrals to account for the constant of integration.
• Combine the power rule with substitution methods like u-substitution for more complex integrals.

In the context of our example, applying the power rule allows us to integrate terms like \(u^{5/2}\) and \(u^{3/2}\). It is crucial for tackling integrals involving polynomials.

table of integrals

A table of integrals is an invaluable resource for quickly finding the antiderivative of a variety of standard functions. These tables list integrals of different forms that frequently occur in calculus, which can save a lot of time when solving integrals by providing solutions to common integration problems.

Using a table of integrals:

• Search for the form of your integrand in the table. Look for patterns that match your problem.
• Follow any additional instructions provided in the table; sometimes, adjustments to the formula are necessary.
• Substitute back any original variables if you used a substitution method before referring to the table.
• Don't forget to append the constant of integration, \(C\), to your final answer.

For the exercise provided, a table of integrals was mentioned initially as a tool that could simplify the problem. However, using u-substitution alongside the power rule was a more direct approach for this specific integral. Integrals tables are particularly useful for complicated expressions and those not easily solved by basic rules, helping you verify solutions and providing guidance when stuck.