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Chapter 5: Problem 15

Find each indefinite integral. \(\int \frac{3 d x}{x}\)

Short Answer

Expert verified
\( 3 \ln|x| + C \)

Step by step solution

01

Identify the form of the integral

The given integral \( \int \frac{3}{x} \, dx \) can be recognized as a form that involves the natural logarithm. Specifically, an integral of the form \( \int \frac{a}{x} \, dx \) results in \( a \ln|x| + C \), where \( a \) is a constant and \( C \) is the constant of integration.
02

Extract the constant

Since the integrand \( \frac{3}{x} \) includes a constant, you can factor it out of the integral: \( 3 \int \frac{1}{x} \, dx \). This expression separates the constant \( 3 \) from the variable-dependent part.
03

Integrate the \(\frac{1}{x}\) portion

Using the basic rule that \( \int \frac{1}{x} \, dx = \ln|x| + C \), compute \( \int \frac{1}{x} \, dx \). Therefore, the integral is equal to \( \ln|x| \).
04

Multiply by the constant

Now, multiply the result from Step 3 by the constant extracted in Step 2: \( 3 \ln|x| \). This gives us the result of the integral.
05

Add the constant of integration

Include the constant of integration \( C \) to account for the family of all antiderivative functions: \( 3 \ln|x| + C \). This is the general solution to the integral.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept encountered in calculus, especially in integration. It is a logarithm with the base of the irrational number \( e \) (approximately 2.71828). The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain \( x \). In calculus, the natural logarithm is deeply intertwined with exponential functions since differentiation and integration involving exponential functions often yield log functions.
  • For any positive \( x \), \( \ln(x) \) provides a relation between the area under the curve of \( y=1/x \) from 1 to \( x \).
  • In the context of integration, \( \int \frac{1}{x} \, dx = \ln|x| + C \) is a fundamental rule, used to find antiderivatives involving \( \frac{1}{x} \).
By understanding this basic integration rule, you can handle a wide variety of integral problems that mirror this form.
Constant of Integration
The constant of integration, symbolized by \( C \), is a crucial part of solving any indefinite integral. An indefinite integral represents a family of functions, each differing by a constant value. This constant accounts for all possible vertical shifts of the antiderivative function that will satisfy a given derivative.
  • When you compute an indefinite integral, like \( \int f(x) \, dx = F(x) + C \), \( C \) represents any arbitrary real number.
  • The inclusion of \( C \) ensures completeness, reflecting the idea that derivatives erase constant terms; thus, differentiation of \( F(x) + C \) would return to the original function \( f(x) \).
Recognizing the necessity of \( C \) helps in understanding that an indefinite integral does not yield one fixed solution but a family of solutions.
Antiderivative
An antiderivative is essentially the reverse of the process of differentiation. If \( F(x) \) is an antiderivative of \( f(x) \), this means that taking the derivative of \( F(x) \) gives you \( f(x) \).
  • Every function can have multiple antiderivatives, each varying by a constant, which leads us to expressions such as \( F(x) + C \).
  • The concept of the antiderivative is crucial for finding solutions to differential equations and solving areas under curves.
In integration problems, identifying an antiderivative involves recognizing patterns in functions and applying known integration techniques, such as recognizing \( \int \frac{1}{x} \, dx \) as \( \ln|x| + C \).
Calculus Integration Techniques
Calculus integration techniques are diverse and designed to approach a wide variety of integrals encountered in mathematical problems. Some main techniques include substitution, integration by parts, and partial fraction decomposition, among others. Each technique is applied based on the form and complexity of the integrand.
  • Substitution is used when an integrand is a composite function that can be transformed into a simpler form by changing the variables.
  • Integration by parts is helpful when the integrand is a product of functions, particularly when combining a polynomial and a log or exponential function.
  • Recognizing the form of the integrand like \( \int \frac{1}{x} \, dx \), easily identifiable forms like these allow the direct application of natural logarithm rule, simplifying calculations.
Mastery of these techniques equips students to tackle complex integrals efficiently.

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Most popular questions from this chapter

Find the derivative of each function. $$ \ln \left(x^{3}+6 x\right) $$

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found in this way (as well as by the methods of Section 6.1). $$ \int(x-2)(x+4)^{5} d x $$

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The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found in this way (as well as by the methods of Section 6.1). $$ \int x(x+4)^{7} d x $$

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