Problem 6
Question
Use the table of integrals at the back of the book to evaluate the integrals in Exercises \(1-26 .\) $$ \int x(7 x+5)^{3 / 2} d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \( \frac{1}{343}(7x+5)^{7/2} - \frac{2}{49}(7x+5)^{5/2} + C \).
1Step 1: Identify the Appropriate Integral Formula
To evaluate the integral \( \int x(7x+5)^{3/2} \, dx \), we first recognize this as a function with the form \( \int x (ax+b)^n \, dx \), where \( a = 7 \), \( b = 5 \), and \( n = \frac{3}{2} \). We need a formula for integrating such functions.
2Step 2: Select a Substitution
To simplify the integration process, let's use substitution. Set \( u = 7x + 5 \). Then \( du = 7 \, dx \) which implies \( dx = \frac{1}{7} \, du \). Also, note that \( x = \frac{u-5}{7} \).
3Step 3: Rewrite the Integral in Terms of u
Substitute \( u \) and \( du \) into the integral: \[ \int x(7x+5)^{3/2} \, dx = \int \frac{u-5}{7} u^{3/2} \, \frac{1}{7} \, du \].This simplifies to:\[ \int \frac{(u-5)u^{3/2}}{49} \, du \].
4Step 4: Simplify the Expression
Expand the expression inside the integral:\[ \int \frac{u^{5/2} - 5u^{3/2}}{49} \, du = \frac{1}{49} \int (u^{5/2} - 5u^{3/2}) \, du \].
5Step 5: Integrate Using Power Rule
Apply the power rule for integration, \( \int u^m \, du = \frac{u^{m+1}}{m+1} + C \):- Integrate \( u^{5/2} \): \( \frac{u^{7/2}}{7/2} = \frac{2}{7} u^{7/2} \).- Integrate \( 5u^{3/2} \): \( 5 \cdot \frac{u^{5/2}}{5/2} = 2u^{5/2} \).Thus the integrated form is:\[ \frac{1}{49} \left( \frac{2}{7} u^{7/2} - 2u^{5/2} \right) + C \].
6Step 6: Substitute Back in Terms of x
Replace \( u \) with \( 7x + 5 \) to return to the variable \( x \):\[ \frac{1}{49} \left( \frac{2}{7} (7x + 5)^{7/2} - 2 (7x+5)^{5/2} \right) + C \].
7Step 7: Simplify the Final Expression
Simplify the constants and expression:\[ \frac{1}{343} (7x+5)^{7/2} - \frac{2}{49} (7x+5)^{5/2} + C \].
8Step 8: Check the Final Answer
Review the steps and ensure the solution is correct. The answer matches common integral tables for solutions involving powers of a linear function of \( x \).
Key Concepts
Integral substitutionPower ruleIntegration using tables of integralsDefinite and indefinite integrals
Integral substitution
Integral substitution is a technique used in calculus to make difficult integrals more manageable. In the original solution, we encounter the integral \( \int x(7x+5)^{3/2} \, dx \). Here, substitution helps simplify the integrand into a familiar form that is easier to integrate.
We begin by identifying a part of the integrand to substitute. The choice \( u = 7x + 5 \) means the derivative of \( u \), or \( du \), is \( 7 \, dx \). This relationship suggests \( dx = \frac{1}{7} \, du \).
This substitution simplifies the process because it transforms a product of terms into a polynomial in terms of \( u \), allowing us to directly apply integration techniques. In essence, substitution is about changing variables to streamline the integration process.
We begin by identifying a part of the integrand to substitute. The choice \( u = 7x + 5 \) means the derivative of \( u \), or \( du \), is \( 7 \, dx \). This relationship suggests \( dx = \frac{1}{7} \, du \).
This substitution simplifies the process because it transforms a product of terms into a polynomial in terms of \( u \), allowing us to directly apply integration techniques. In essence, substitution is about changing variables to streamline the integration process.
Power rule
The power rule is a straightforward method for integrating functions involving powers of variables. It is expressed as: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \], where \( n eq -1 \), and \( C \) is the constant of integration.
In the given integral, after substitution, we have \( u^{5/2} \) and \( 5u^{3/2} \). We apply the power rule to each separately: - \( \int u^{5/2} \, du = \frac{u^{7/2}}{7/2} + C \) - \( \int 5u^{3/2} \, du = 2u^{5/2} + C \)
The power rule simplifies these expressions by applying the general formula for integration, resulting in straightforward answers for each term. Using the power rule effectively allows one to manage polynomial integrands with power functions.
In the given integral, after substitution, we have \( u^{5/2} \) and \( 5u^{3/2} \). We apply the power rule to each separately: - \( \int u^{5/2} \, du = \frac{u^{7/2}}{7/2} + C \) - \( \int 5u^{3/2} \, du = 2u^{5/2} + C \)
The power rule simplifies these expressions by applying the general formula for integration, resulting in straightforward answers for each term. Using the power rule effectively allows one to manage polynomial integrands with power functions.
Integration using tables of integrals
Tables of integrals are valuable resources for quickly finding integral solutions that might be cumbersome to derive manually. These tables list common integral formulas that can be applied directly to problems with matching forms.
When approaching the integral \( \int x(7x+5)^{3/2} \, dx \), it's identified to follow a standard form from the tables, particularly \( \int x (ax+b)^n \, dx \). By recognizing this form, direct references to integral tables help confirm calculations or provide a starting point for more complex manipulations.
Using tables simplifies work and ensures accuracy, especially with less common or more complicated integral forms, minimizing calculation errors and saving time.
When approaching the integral \( \int x(7x+5)^{3/2} \, dx \), it's identified to follow a standard form from the tables, particularly \( \int x (ax+b)^n \, dx \). By recognizing this form, direct references to integral tables help confirm calculations or provide a starting point for more complex manipulations.
Using tables simplifies work and ensures accuracy, especially with less common or more complicated integral forms, minimizing calculation errors and saving time.
Definite and indefinite integrals
Integrals can either be definite or indefinite, depending on whether they represent an accumulation over an interval or an antiderivative of a function. An indefinite integral is expressed as \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative and \( C \) is the constant of integration.
The problem presented is an indefinite integral, aiming to find the general antiderivative of the function \( x(7x+5)^{3/2} \).
Indefinite integrals are used to define a family of functions as solutions, characterized by the constant \( C \). Unlike definite integrals, which calculate a specific numerical answer over a range, indefinite integrals offer broader solutions applicable to a wide variety of contexts.
The problem presented is an indefinite integral, aiming to find the general antiderivative of the function \( x(7x+5)^{3/2} \).
Indefinite integrals are used to define a family of functions as solutions, characterized by the constant \( C \). Unlike definite integrals, which calculate a specific numerical answer over a range, indefinite integrals offer broader solutions applicable to a wide variety of contexts.
Other exercises in this chapter
Problem 5
The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropr
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The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule
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The integrals in Exercises \(1-34\) converge. Evaluate the integrals without using tables. $$\int_{-8}^{1} \frac{d x}{x^{1 / 3}}$$
View solution Problem 6
Evaluate the integrals in Exercises \(1-14\) $$ \int_{0}^{1 / 2 \sqrt{2}} \frac{2 d x}{\sqrt{1-4 x^{2}}} $$
View solution