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Magazine Chapter 6: Problem 20
Evaluate the definite integrals. \(\int_{0}^{4} \frac{1}{\sqrt{2 x+1}} d x\)
Short Answer
Expert verified
The integral evaluates to 2.
Step by step solution
01
Identify the Function and Limits
The given integral is \( \int_{0}^{4} \frac{1}{\sqrt{2x+1}} \, dx \). The limits of integration are from \( x = 0 \) to \( x = 4 \). We need to find an antiderivative of the function \( \frac{1}{\sqrt{2x+1}} \).
02
Perform Substitution
To simplify the integration process, we use substitution. Let \( u = 2x + 1 \). Then the derivative \( du = 2 \, dx \) or \( dx = \frac{1}{2} \, du \). Change the limits of integration accordingly: when \( x = 0 \), \( u = 1 \); when \( x = 4 \), \( u = 9 \). Substitute into the integral: \( \int_{1}^{9} \frac{1}{\sqrt{u}} \cdot \frac{1}{2} \, du \).
03
Integrate the New Function
The integral now is \( \frac{1}{2} \int_{1}^{9} u^{-1/2} \, du \). Find the antiderivative: \( \int u^{-1/2} \, du = 2u^{1/2} + C \). Thus, the integral is \( \frac{1}{2} \cdot 2u^{1/2} \) or \( u^{1/2} \).
04
Evaluate the Definite Integral
Substitute the limits back into the antiderivative: \( [u^{1/2}]_{1}^{9} \). Compute: \( (9)^{1/2} - (1)^{1/2} = 3 - 1 = 2 \).
05
Conclude the Solution
Therefore, the value of the definite integral \( \int_{0}^{4} \frac{1}{\sqrt{2x+1}} \, dx \) is 2.
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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 to simplify the process of finding integrals, especially when dealing with complex functions. In integration, substitution is similar to the chain rule used in differentiation.
**How it Works**:
- We make a substitution to change the variable of integration to a new variable that simplifies the integral.
- Choose a substitution that makes the inner part of the composite function simpler.
- Express the original variable of integration in terms of the new variable and adjust the differential accordingly.
- Sometimes, the integral limits need adjustment when switching variables, which requires recalculating limits for the new variable.
In our example, we substitute by letting \( u = 2x + 1 \). This choice simplifies the radical expression, making integration manageable. Derive \( du = 2 \, dx \), which modifies \( dx \) to \( \frac{1}{2} du \), and adjust the integration limits as needed. From \( x = 0 \) to \( x = 4 \), the values of \( u \) change from 1 to 9. With this approach, finding the antiderivative becomes more straightforward.
**How it Works**:
- We make a substitution to change the variable of integration to a new variable that simplifies the integral.
- Choose a substitution that makes the inner part of the composite function simpler.
- Express the original variable of integration in terms of the new variable and adjust the differential accordingly.
- Sometimes, the integral limits need adjustment when switching variables, which requires recalculating limits for the new variable.
In our example, we substitute by letting \( u = 2x + 1 \). This choice simplifies the radical expression, making integration manageable. Derive \( du = 2 \, dx \), which modifies \( dx \) to \( \frac{1}{2} du \), and adjust the integration limits as needed. From \( x = 0 \) to \( x = 4 \), the values of \( u \) change from 1 to 9. With this approach, finding the antiderivative becomes more straightforward.
Antiderivatives
Finding antiderivatives is crucial in solving integrals. An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that the derivative \( F'(x) = f(x) \).
**The Process:**
- Determine the form of the antiderivative; it's often based on known formulas or rules of integral calculus.
- Always add a constant \( C \) when dealing with indefinite integrals, as integration is the inverse operation of differentiation.
In the given exercise, we look for an antiderivative for \( u^{-1/2} \). The result is \( 2u^{1/2} \), and the process involves integrating the function \( u^{-1/2} \), simplifying to \( 2u^{1/2} + C \) for an indefinite integral. However, since we're dealing with a definite integral, the limits will evaluate the constant away, allowing us to focus purely on the evaluation from the new limits of \( u \). The conversion from the indefinite result is \( \frac{1}{2} \cdot 2u^{1/2} \) simplifies ultimately to \( u^{1/2} \) without the constant in definite cases.
**The Process:**
- Determine the form of the antiderivative; it's often based on known formulas or rules of integral calculus.
- Always add a constant \( C \) when dealing with indefinite integrals, as integration is the inverse operation of differentiation.
In the given exercise, we look for an antiderivative for \( u^{-1/2} \). The result is \( 2u^{1/2} \), and the process involves integrating the function \( u^{-1/2} \), simplifying to \( 2u^{1/2} + C \) for an indefinite integral. However, since we're dealing with a definite integral, the limits will evaluate the constant away, allowing us to focus purely on the evaluation from the new limits of \( u \). The conversion from the indefinite result is \( \frac{1}{2} \cdot 2u^{1/2} \) simplifies ultimately to \( u^{1/2} \) without the constant in definite cases.
Limits of Integration
In definite integrals, limits of integration play a critical role. They define the region over which the integration occurs, binding the evaluation of the function to precise points on the axis.
**Changing Limits with Substitution:**
- When substitution is used, original limits (\( x \) values) must be converted to limits in terms of the new variable (\( u \) values).
- This involves substituting the original limits into the substitution formula.
In our example, the integral begins with limits from \( x = 0 \) to \( x = 4 \). After substitution, these limits change based on \( u = 2x + 1 \):
\[ \text{When } x = 0, \, u = 2(0) + 1 = 1. \]
\[ \text{When } x = 4, \, u = 2(4) + 1 = 9. \]
This clear conversion helps us accurately evaluate the definite integral from \( u = 1 \) to \( u = 9 \). This step is crucial because correct limits ensure the integral value truly represents the function鈥檚 area over the specified range.
**Changing Limits with Substitution:**
- When substitution is used, original limits (\( x \) values) must be converted to limits in terms of the new variable (\( u \) values).
- This involves substituting the original limits into the substitution formula.
In our example, the integral begins with limits from \( x = 0 \) to \( x = 4 \). After substitution, these limits change based on \( u = 2x + 1 \):
\[ \text{When } x = 0, \, u = 2(0) + 1 = 1. \]
\[ \text{When } x = 4, \, u = 2(4) + 1 = 9. \]
This clear conversion helps us accurately evaluate the definite integral from \( u = 1 \) to \( u = 9 \). This step is crucial because correct limits ensure the integral value truly represents the function鈥檚 area over the specified range.
