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Magazine Chapter 6: Problem 54
Evaluate the integrals. $$ \int_{0}^{1}\left(1-e^{x}\right)^{10} e^{x} d x $$
Short Answer
Expert verified
\( \frac{(e-1)^{11}}{11} \)
Step by step solution
01
Recognize the Integral Type
Identify that the integral \( \int_{0}^{1} (1-e^{x})^{10} e^{x} \, dx \) is a definite integral with limits from 0 to 1.
02
Substitution Method
Let us use substitution to simplify the integral. Set \( u = 1 - e^{x} \). Then \( du = -e^{x} \, dx \), so \( e^{x} \, dx = -du \).
03
Change the Limits of Integration
With the substitution \( u = 1 - e^{x} \), change the limits of integration: when \( x = 0 \), \( u = 1 - e^0 = 0 \); when \( x = 1 \), \( u = 1 - e^1 = 1-e \).
04
Rewrite the Integral
Using the substitution, the integral becomes:\[ -\int_{0}^{1-e} u^{10} \, du \]
05
Evaluate the Integral
Integrate \( -\int_{0}^{1-e} u^{10} \, du \). The antiderivative of \( u^{10} \) is \( \frac{u^{11}}{11} \). Substituting limits,\[ -\left[ \frac{u^{11}}{11} \right]_{0}^{1-e} \]
06
Calculate the Result
Compute \(-\left[ \frac{(1-e)^{11}}{11} - \frac{0^{11}}{11} \right]\ = \frac{(e-1)^{11}}{11} \).
07
Simplify the Result
Since \((e-1)^{11}\) in absolute value is the symmetrical value to \((1-e)^{11}\) but positive, thus,\[ \frac{(e-1)^{11}}{11} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique used in calculus to simplify complex integrals. In this method, we transform the original variable into a new variable, making the integral easier to solve. For the given problem, we initially have \( \int (1-e^x)^{10} e^x \, dx \).
- We identify the substitution as \( u = 1 - e^x \). This is because the presence of \( (1-e^x)^{10} \) and \( e^x \) hints at a function and its derivative inside the integral.
- Calculating the derivative gives us \( du = -e^x \, dx \) or \( e^x \, dx = -du \).
- This substitution effectively transforms the integral from a complex form into a polynomial, \( \int u^{10} \, du \), which is simpler to integrate.
Limits of Integration
When using the substitution method, adjusting the limits of integration is crucial. The original integral has limits for \( x \) ranging from 0 to 1. However, with substitution \( u = 1 - e^x \), these limits must be converted:
- At \( x = 0 \), substitute to find \( u = 1 - e^0 = 0 \).
- At \( x = 1 \), substitute to find \( u = 1 - e^1 = 1-e \).
Antiderivative
Finding the antiderivative is the core step in solving integrals. For our transformed integral \( \int u^{10} \, du \), we need to find its antiderivative. This involves:
- Recognizing the function \( u^{10} \) as a polynomial, which integrates straightforwardly using the formula \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).
- Applying this to get the antiderivative, \( \frac{u^{11}}{11} \), which represents the indefinite integral of \( u^{10} \).
Evaluate Integrals
After obtaining the antiderivative, the final challenge is evaluating the integral using the new limits. For the integral \( -\int_{0}^{1-e} u^{10} \, du \), this process involves:
- Inserting the upper limit \( u = 1-e \) into the antiderivative, giving \( \frac{(1-e)^{11}}{11} \).
- Inserting the lower limit \( u = 0 \), which results in \( \frac{0^{11}}{11} = 0 \).
- Calculating the difference, which in this case is \( -\left( \frac{(1-e)^{11}}{11} - 0 \right) \).
