Problem 19
Question
Evaluate the integral. \(\int_{0}^{1}\left(x^{e}+e^{x}\right) d x\)
Step-by-Step Solution
Verified Answer
The integral evaluates to \(\frac{1}{e+1} + (e - 1)\).
1Step 1: Identify the Integral
The given exercise asks us to evaluate the definite integral \(\int_{0}^{1}\left(x^{e}+e^{x}\right) d x\) from 0 to 1. This integral is the sum of two separate integrals: \(\int_{0}^{1} x^{e} \, dx\) and \(\int_{0}^{1} e^{x} \, dx\).
2Step 2: Evaluate the First Integral
To find \(\int_{0}^{1} x^{e} \, dx\), we use the power rule for integration. The power rule states:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]For \(x^{e}\), the result is \(\frac{x^{e+1}}{e+1}\). Evaluating from 0 to 1:\[ \left[\frac{x^{e+1}}{e+1}\right]_{0}^{1} = \frac{1^{e+1}}{e+1} - \frac{0^{e+1}}{e+1} = \frac{1}{e+1} \]
3Step 3: Evaluate the Second Integral
For \(\int_{0}^{1} e^{x} \, dx\), we remember the rule for integrating \(e^{x}\), which is:\[ \int e^x \, dx = e^x + C \]So, evaluate from 0 to 1:\[ \left[e^x\right]_{0}^{1} = e^1 - e^0 = e - 1 \]
4Step 4: Add the Results
Now we sum the results of the integrals:\[ \int_{0}^{1} \, (x^{e} + e^{x}) \, dx = \frac{1}{e+1} + (e - 1) \]This gives the final result of the definite integral.
Key Concepts
Power Rule in IntegrationExponential Function IntegrationCalculus Problem Solving Steps
Power Rule in Integration
The Power Rule in Integration is a cornerstone of calculus that streamlines the integration process when dealing with polynomial functions. It's quite similar to the power rule used in differentiation, making it a familiar concept to many students.
This rule specifically states that for any real number \( n \), the integral of \( x^n \) with respect to \( x \) is:
In our scenario, for the integral \( \int x^e \, dx \), you can substitute \( n \) with the constant \( e \), Euler's number, which is approximately 2.718. Applying the power rule results in \( \frac{x^{e+1}}{e+1} \).
Evaluating this definite integral from 0 to 1 involved plugging these limits into the integrated function which led us to \( \frac{1}{e+1} \). Through the proper application of the power rule, tackling integrals of this form becomes straightforward and manageable.
This rule specifically states that for any real number \( n \), the integral of \( x^n \) with respect to \( x \) is:
- \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
In our scenario, for the integral \( \int x^e \, dx \), you can substitute \( n \) with the constant \( e \), Euler's number, which is approximately 2.718. Applying the power rule results in \( \frac{x^{e+1}}{e+1} \).
Evaluating this definite integral from 0 to 1 involved plugging these limits into the integrated function which led us to \( \frac{1}{e+1} \). Through the proper application of the power rule, tackling integrals of this form becomes straightforward and manageable.
Exponential Function Integration
Integrating the exponential function \( e^x \) is a fundamental process in calculus that many students will encounter. Exponential functions have a unique property in integration and differentiation. The integral of \( e^x \) respects the form of the function.
In our current example, we applied this property over the interval from 0 to 1, leading to the expression \( [e^x]_{0}^{1} \). Evaluating this from the lower to upper limit gives us \( e - 1 \). Knowing how to integrate exponential functions smoothly and efficiently is crucial, as they frequently appear in calculus problems.
- The rule to remember is: \( \int e^x \, dx = e^x + C \).
In our current example, we applied this property over the interval from 0 to 1, leading to the expression \( [e^x]_{0}^{1} \). Evaluating this from the lower to upper limit gives us \( e - 1 \). Knowing how to integrate exponential functions smoothly and efficiently is crucial, as they frequently appear in calculus problems.
Calculus Problem Solving Steps
Solving calculus problems, especially those involving integration, often requires a structured approach to ensure accuracy and comprehension. Here are some steps to guide you through such problems:
1. **Identify the Integral**
Determine what type of integral you are dealing with. Is it definite or indefinite? Identifying the integral helps outline the approach you need to take.2. **Simplify where Possible**
Break complex expressions into simpler parts if needed. In our problem, this involved separating \( \int (x^e + e^x) \, dx \) into two integral parts.3. **Apply the Rules**
Use the appropriate integration rules. For polynomial terms use the power rule, and for exponential terms use the exponential integration rules.4. **Evaluate the Integral**
For definite integrals, calculate the result over the specified interval by substituting the limits.5. **Verify Your Result**
Cross-check your solution and ensure it follows logical steps and accurate arithmetic.
By following these steps, you maintain order in solving calculus problems and reduce the chance of errors, much like we demonstrated with solving the integral \( \int_{0}^{1}(x^e + e^x) \, dx \). Every problem might demand its unique approach, but generally, these steps provide a solid framework.
1. **Identify the Integral**
Determine what type of integral you are dealing with. Is it definite or indefinite? Identifying the integral helps outline the approach you need to take.2. **Simplify where Possible**
Break complex expressions into simpler parts if needed. In our problem, this involved separating \( \int (x^e + e^x) \, dx \) into two integral parts.3. **Apply the Rules**
Use the appropriate integration rules. For polynomial terms use the power rule, and for exponential terms use the exponential integration rules.4. **Evaluate the Integral**
For definite integrals, calculate the result over the specified interval by substituting the limits.5. **Verify Your Result**
Cross-check your solution and ensure it follows logical steps and accurate arithmetic.
By following these steps, you maintain order in solving calculus problems and reduce the chance of errors, much like we demonstrated with solving the integral \( \int_{0}^{1}(x^e + e^x) \, dx \). Every problem might demand its unique approach, but generally, these steps provide a solid framework.
Other exercises in this chapter
Problem 19
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