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Magazine Chapter 7: Problem 9
Evaluate. $$ \int x(1+x)^{10} d x $$
Short Answer
Expert verified
The integral evaluates to \( \frac{(1+x)^{12}}{12} - \frac{(1+x)^{11}}{11} + C \).
Step by step solution
01
Set up the Integral
We have the integral \( \int x(1+x)^{10} \, dx \). This is a basic integral where the function inside is a polynomial multiplied by another function. We will need to apply integration techniques to solve it.
02
Use Substitution Method
To solve this integral, we should use substitution. Let \( u = 1 + x \) such that \( du = dx \). When \( x = 0, u = 1 \). Hence \( x = u - 1 \). Substitute these into the integral.
03
Substitute Variables
Replace \( x \) and \( dx \) in the integral: \( \int x(1+x)^{10} \, dx = \int (u - 1)u^{10} \, du \). This simplifies the integral.
04
Expand the Expression
Expand \( (u - 1)u^{10} \) to \( u^{11} - u^{10} \). Now, the integral becomes \( \int (u^{11} - u^{10}) \, du \).
05
Integrate Each Term Separately
Integrate term by term: \( \int u^{11} \, du = \frac{u^{12}}{12} \) and \( \int u^{10} \, du = \frac{u^{11}}{11} \).
06
Combine the Integrals
Combine the integrals to get \( \frac{u^{12}}{12} - \frac{u^{11}}{11} + C \), where \( C \) is the constant of integration.
07
Substitute Back the Original Variable
Since \( u = 1 + x \), substitute back: \( \frac{(1+x)^{12}}{12} - \frac{(1+x)^{11}}{11} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
When facing a complex integral like \( \int x(1+x)^{10} \, dx \), the substitution method can simplify the process. Think of substitution as a smart change of variables to make the integral more manageable.
Here's how it works in this scenario:
Here's how it works in this scenario:
- Choose a substitution that simplifies the inner part of your function. Here, we set \( u = 1 + x \).
- Differentiate your new variable to find \( du \). Since \( du = dx \), it lines up perfectly with the integral components.
- Replace the relevant parts in the integral. For our substitution, \( x = u - 1 \), leading us to transform the integral from \( x(1+x)^{10} \, dx \) to \( (u-1)u^{10} \, du \).
Polynomial Integration
Polynomial integration involves integrating terms raised to a power, such as \( u^{11} \) or \( u^{10} \). After substitution in our integral, we expand \( (u-1)u^{10} \) to \( u^{11} - u^{10} \). This is key because integrating polynomials is straightforward.
For each term of the polynomial:
For each term of the polynomial:
- Integrate separately: Break down the expression to tackle each term one-by-one. For example, integrate \( u^{11} \) and \( u^{10} \) individually.
- Use the power rule for integration: The power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). So, \( \int u^{11} = \frac{u^{12}}{12} \) and \( \int u^{10} = \frac{u^{11}}{11} \).
- Combine the results: After integrating, organize all the terms together with a constant of integration \( C \).
Definite Integrals
While our original exercise focused on finding an indefinite integral, understanding definite integrals is crucial as it allows us to evaluate the exact area under a curve over a specific interval. In a definite integral, the function is bounded by limits, usually indicated by an expression like \( \int_{a}^{b} f(x) \, dx \). Here, \( a \) and \( b \) are the limits of integration.
Here are key aspects:
Here are key aspects:
- Limits of integration: These limits narrow the focus of the integral to a specific section of the function curve.
- Apply the substitution technique: Just like with indefinite integrals, substitutions can simplify expressions, making it easier to evaluate between limits.
- Calculate boundary values: Once the indefinite integral is found, apply the limits by substituting them into the anti-derivative function. Evaluate the result by finding the difference between these values.
