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Magazine Chapter 3: Problem 244
Use a table of integrals to evaluate the following integrals. $$ \int_{0}^{4} \frac{x}{\sqrt{1+2 x}} d x $$
Short Answer
Expert verified
The value of the integral is \( \frac{10}{3} \).
Step by step solution
01
Identify the Integral Form
We begin by looking at the given integral \( \int_{0}^{4} \frac{x}{\sqrt{1+2x}} \, dx \). We need to find a similar form in the table of integrals to make use of standard results or substitution methods for evaluation.
02
Use Substitution
Notice that if we let \( u = 1 + 2x \), then \( du = 2 \, dx \) or equivalently, \( dx = \frac{du}{2} \). Also, when \( x = 0 \), \( u = 1 \), and when \( x = 4 \), \( u = 9 \). Substitute these into the integral.
03
Adjust the Integral with Substitution
With the substitution \( u = 1 + 2x \), \( x = \frac{u - 1}{2} \), and \( dx = \frac{du}{2} \), the integral becomes:\[ \int_{1}^{9} \frac{\frac{u-1}{2}}{\sqrt{u}} \cdot \frac{du}{2} = \frac{1}{4} \int_{1}^{9} \frac{u - 1}{\sqrt{u}} \, du \].
04
Simplify the Integral
Simplify the expression \( \frac{u - 1}{\sqrt{u}} \) as \( \sqrt{u} - \frac{1}{\sqrt{u}} \). This breaks the integral into two separate integrals:\[ \frac{1}{4} \left( \int_{1}^{9} u^{1/2} \, du - \int_{1}^{9} u^{-1/2} \, du \right) \].
05
Integrate Each Term Separately
Now, apply the power rule of integration to each term:- For \( \int u^{1/2} \, du \), the integral is \( \frac{2}{3} u^{3/2} \).- For \( \int u^{-1/2} \, du \), the integral is \( 2u^{1/2} \).Then the entire integral becomes:\[ \frac{1}{4} \left( \left. \frac{2}{3} u^{3/2} \right|_1^9 - \left. 2u^{1/2} \right|_1^9 \right) \].
06
Evaluate the Definite Integrals
Compute the values at the bounds:- \( \left. \frac{2}{3} u^{3/2} \right|_1^9 = \frac{2}{3} (9^{3/2} - 1^{3/2}) = \frac{2}{3} (27 - 1) = \frac{2}{3} \times 26 = \frac{52}{3} \).- \( \left. 2u^{1/2} \right|_1^9 = 2(9^{1/2} - 1^{1/2}) = 2(3 - 1) = 4 \).Hence, combine them: \[ \frac{1}{4} \left( \frac{52}{3} - 4 \right) \].
07
Final Calculation
Calculate the final expression for the definite integral:- Simplify the expression inside the brackets: \( \frac{52}{3} - 4 = \frac{52}{3} - \frac{12}{3} = \frac{40}{3} \).- Multiply by \( \frac{1}{4} \):\[ \frac{1}{4} \times \frac{40}{3} = \frac{40}{12} = \frac{10}{3} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Understanding definite integrals is a fundamental part of integral calculus. When you compute a definite integral, you are essentially finding the accumulated value, which represents the area under a curve, from one point to another on a graph. Definite integrals have both an upper and lower limit; in this case, from 0 to 4.
Definite integrals are represented as \( \int_{a}^{b} f(x) \, dx \), with \( a \) and \( b \) being the limits of integration. After evaluating the integral function from \( a \) to \( b \), you subtract its value at \( a \) from its value at \( b \).
Definite integrals are represented as \( \int_{a}^{b} f(x) \, dx \), with \( a \) and \( b \) being the limits of integration. After evaluating the integral function from \( a \) to \( b \), you subtract its value at \( a \) from its value at \( b \).
- Integrating functions can help find the total accumulation of a changing quantity.
- It's crucial to pay attention to the limits provided as they affect the outcome.
Substitution Method
The substitution method, also known as the \( u \)-substitution technique, is an essential tool in integral calculus. It helps simplify complex integrals by transforming them into a more manageable form.
Here's how it generally works:
Here's how it generally works:
- You choose a substitution, usually a part of the integral, to represent with a new variable, \( u \).
- Express the differential \( dx \) in terms of \( du \).
- Change the limits of integration according to your substitution.
Power Rule of Integration
The power rule of integration is a key technique for integrating terms involving powers of a variable. It states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \), and \( C \) is the constant of integration for indefinite integrals.
The exercise showcases the application of this rule by transforming the integral into terms that are powers of \( u \). At this stage, the original integral has been broken down into more basic components using substitution, and the power rule is applied:
The exercise showcases the application of this rule by transforming the integral into terms that are powers of \( u \). At this stage, the original integral has been broken down into more basic components using substitution, and the power rule is applied:
- The term \( u^{1/2} \) is integrated as \( \frac{2}{3} u^{3/2} \).
- The term \( u^{-1/2} \) is integrated as \( 2u^{1/2} \).
