Problem 33
Question
Find each integral. $$ \int 6 e^{8 x} d x $$
Step-by-Step Solution
Verified Answer
The integral is \( \frac{3}{4} e^{8x} + C \).
1Step 1: Identifying the Integral Type
The integral given is \( \int 6 e^{8x} \, dx \). Notice that this is an integral of the form \( \int a e^{bx} \, dx \), where \( a \) and \( b \) are constants. This is a standard exponential integral, which suggests the use of a basic formula for exponential functions.
2Step 2: Applying the Exponential Rule
For an integral of the form \( \int a e^{bx} \, dx \), the integral is solved using the formula: \( \int a e^{bx} \, dx = \frac{a}{b} e^{bx} + C \), where \( C \) is the constant of integration.
3Step 3: Substituting Values into the Formula
In our case, \( a = 6 \) and \( b = 8 \). Applying these values into the formula gives: \( \int 6 e^{8x} \, dx = \frac{6}{8} e^{8x} + C \).
4Step 4: Simplifying the Expression
Simplify the expression \( \frac{6}{8} \) to \( \frac{3}{4} \) to get \( \int 6 e^{8x} \, dx = \frac{3}{4} e^{8x} + C \).
Key Concepts
Definite and Indefinite IntegralsExponential FunctionsIntegration Techniques
Definite and Indefinite Integrals
When we talk about integrals in calculus, we often deal with two types: definite and indefinite integrals. Understanding the difference between these is crucial for solving any integration problem successfully.
An **indefinite integral** represents a family of functions and includes a constant of integration, denoted as \( C \). This is what you find when you are asked to calculate \( \int f(x) \, dx \). The output is essentially a function, plus this constant \( C \) that accounts for all possible vertical shifts of the antiderivative.
On the other hand, a **definite integral** has limits of integration – upper and lower bounds – and it calculates the net area under the curve of a function between these two points. As such, definite integrals evaluate to numbers rather than functions. They are denoted by \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration.
In the exercise **\( \int 6 e^{8x} \, dx \)**, we are dealing with an indefinite integral, as there are no limits specified. Here, the solution involves finding a general expression for the antiderivative of the function \( 6 e^{8x} \) and includes a constant of integration \( C \). This means the solution is a family of functions rather than a numeric value.
An **indefinite integral** represents a family of functions and includes a constant of integration, denoted as \( C \). This is what you find when you are asked to calculate \( \int f(x) \, dx \). The output is essentially a function, plus this constant \( C \) that accounts for all possible vertical shifts of the antiderivative.
On the other hand, a **definite integral** has limits of integration – upper and lower bounds – and it calculates the net area under the curve of a function between these two points. As such, definite integrals evaluate to numbers rather than functions. They are denoted by \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration.
In the exercise **\( \int 6 e^{8x} \, dx \)**, we are dealing with an indefinite integral, as there are no limits specified. Here, the solution involves finding a general expression for the antiderivative of the function \( 6 e^{8x} \) and includes a constant of integration \( C \). This means the solution is a family of functions rather than a numeric value.
Exponential Functions
Exponential functions are a key concept in mathematics and appear frequently in calculus problems, particularly in integration and differentiation. These functions are of the form \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of natural logarithms, approximately equal to 2.718.
These functions grow very quickly because, for every increase in \( x \), the function's value is multiplied by \( e^{b} \). Consequently, exponential functions are used to model many natural processes, such as growth and decay, population dynamics, and even financial calculations like compound interest.
In our given exercise, \( 6 e^{8x} \) is an exponential function with constants \( a = 6 \) and \( b = 8 \). This function describes a rapidly increasing quantity due to the positive exponent, and our integration task involves finding the general formula for the accumulation of this growth over time.
These functions grow very quickly because, for every increase in \( x \), the function's value is multiplied by \( e^{b} \). Consequently, exponential functions are used to model many natural processes, such as growth and decay, population dynamics, and even financial calculations like compound interest.
In our given exercise, \( 6 e^{8x} \) is an exponential function with constants \( a = 6 \) and \( b = 8 \). This function describes a rapidly increasing quantity due to the positive exponent, and our integration task involves finding the general formula for the accumulation of this growth over time.
Integration Techniques
Learning different integration techniques is essential to mastering calculus, as they provide various methods to solve integrals effectively. The choice of technique often depends on the form of the function to be integrated.
One fundamental technique involves the use of direct integration formulas, especially for exponential functions. For any integral of the form \( \int a e^{bx} \, dx \), we use the rule:
In the solved exercise **\( \int 6 e^{8x} \, dx \)**, we applied this direct integration rule. By recognizing that \( a = 6 \) and \( b = 8 \), and substituting these values in, we obtained \( \frac{6}{8} e^{8x} + C \), which simplifies to \( \frac{3}{4} e^{8x} + C \).
This integration technique, tailored to exponential functions, allows for an efficient calculation of the integral, making it straightforward and adaptable for more complex expressions involving exponential terms.
One fundamental technique involves the use of direct integration formulas, especially for exponential functions. For any integral of the form \( \int a e^{bx} \, dx \), we use the rule:
- \( \int a e^{bx} \, dx = \frac{a}{b} e^{bx} + C \)
In the solved exercise **\( \int 6 e^{8x} \, dx \)**, we applied this direct integration rule. By recognizing that \( a = 6 \) and \( b = 8 \), and substituting these values in, we obtained \( \frac{6}{8} e^{8x} + C \), which simplifies to \( \frac{3}{4} e^{8x} + C \).
This integration technique, tailored to exponential functions, allows for an efficient calculation of the integral, making it straightforward and adaptable for more complex expressions involving exponential terms.
Other exercises in this chapter
Problem 33
Evaluate using integration by parts. $$ \int_{2}^{0} \ln (x+8) d x $$
View solution Problem 33
Find the area of the region bounded by the graphs of the given equations. $$ y=2 x^{2}-6 x+5, y=x^{2}+6 x-15 $$
View solution Problem 34
Use geometry to evaluate each definite integral. \(\int_{0}^{5} 4 x d x\)
View solution Problem 34
Evaluate. (Be sure to check by differentiating!) $$ \int \frac{d x}{x \ln x^{2}}, x>1 $$
View solution