Problem 57
Question
Find the indefinite integrals. $$\int 5 e^{z} d z$$
Step-by-Step Solution
Verified Answer
\( 5 e^z + C \)
1Step 1: Identify the Form of the Integral
The given integral is \( \int 5 e^z \, dz \). Notice that it is a basic exponential integral of the form \( \int a e^z \, dz \).
2Step 2: Apply the Integral Rule for Exponentials
The rule for integrating \( e^z \) is: \( \int e^z \, dz = e^z + C \), where \( C \) is the constant of integration.
3Step 3: Apply the Constant Multiple Rule
Since there is a constant 5 multiplied with \( e^z \), according to the constant multiple rule, it can be factored out of the integral: \( \int 5 e^z \, dz = 5 \int e^z \, dz \).
4Step 4: Integrate the Expression
Now integrate \( e^z \) according to the rule: \( 5 \int e^z \, dz = 5 (e^z + C) = 5 e^z + C \).
5Step 5: State the Final Result
Thus, the indefinite integral of \( 5 e^z \) with respect to \( z \) is \( 5 e^z + C \).
Key Concepts
Exponential FunctionsIntegration TechniquesConstant of Integration
Exponential Functions
Exponential functions are a type of mathematical function that has the form \( e^x \) where \( e \) is the base of natural logarithms, approximately equal to 2.71828. They are uniquely characterized by the rate of growth being proportional to its current value. This property makes exponential functions incredibly powerful and widely used in various fields such as finance, physics, and biology.
Recognizing exponential functions within integrals is particularly important. The simplicity of \( e^x \) allows us to easily differentiate and integrate it. Its derivative and integral both return the function itself, which can simplify calculations significantly in calculus.
When working with integrals involving exponential functions, it's crucial to remember that the exponential remains unchanged through the process. This makes it an exceptional function as usually, other functions alter significantly during differentiation or integration.
Recognizing exponential functions within integrals is particularly important. The simplicity of \( e^x \) allows us to easily differentiate and integrate it. Its derivative and integral both return the function itself, which can simplify calculations significantly in calculus.
When working with integrals involving exponential functions, it's crucial to remember that the exponential remains unchanged through the process. This makes it an exceptional function as usually, other functions alter significantly during differentiation or integration.
Integration Techniques
Integrating functions involves finding a function whose derivative is the original function you started with. One technique is recognizing the type of function you are working with. For example, with exponential functions, we use the standard rule:\[ \int e^z \, dz = e^z + C.\]
Another technique is applying the constant multiple rule, which simplifies the integration when a function is multiplied by a constant. If you have a constant \( a \) times a function \( f(x) \), you can factor the constant out of the integral, as shown in this formula: \(\int a f(x) \, dx = a \int f(x) \, dx. \)
This helps streamline the process, making it easier to integrate complex expressions. Practicing these techniques strengthens overall calculus skills and is crucial for solving various mathematical problems.
Another technique is applying the constant multiple rule, which simplifies the integration when a function is multiplied by a constant. If you have a constant \( a \) times a function \( f(x) \), you can factor the constant out of the integral, as shown in this formula: \(\int a f(x) \, dx = a \int f(x) \, dx. \)
This helps streamline the process, making it easier to integrate complex expressions. Practicing these techniques strengthens overall calculus skills and is crucial for solving various mathematical problems.
Constant of Integration
In indefinite integrals, the constant of integration, symbolized by \( C \), plays a vital role. When we integrate a function, we add \( C \) because we are looking for all antiderivatives of that function. Rather than a single function, the integral represents a family of functions that differ only by a constant.
For instance, when differentiating the function \( F(x) = x^2 + C \), the derivative \( F'(x) \) equals \( 2x \), regardless of the value \( C \) takes. Adding the constant \( C \) ensures that every possible antiderivative is accounted for when integrating. This is vital in capturing the solution accurately in calculus.
Remember, the constant of integration cannot be omitted, as it represents the infinite set of solutions available in indefinite integration.
For instance, when differentiating the function \( F(x) = x^2 + C \), the derivative \( F'(x) \) equals \( 2x \), regardless of the value \( C \) takes. Adding the constant \( C \) ensures that every possible antiderivative is accounted for when integrating. This is vital in capturing the solution accurately in calculus.
Remember, the constant of integration cannot be omitted, as it represents the infinite set of solutions available in indefinite integration.
Other exercises in this chapter
Problem 56
Find the indefinite integrals. $$\int\left(t^{2}+5 t+1\right) d t$$
View solution Problem 57
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals. $$\int_{0}^{1} 2 t e^{-t^{2}} d t$$
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Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals. $$\int_{-1}^{2} \sqrt{x+2} d x$$
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Find the indefinite integrals. $$\int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t$$
View solution