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Magazine Chapter 28: Problem 10
Integrate each of the given functions. $$\int \frac{x d x}{(x+2)^{4}}$$
Short Answer
Expert verified
The integral is \(-\frac{1}{2(x+2)^2} + \frac{2}{3(x+2)^3} + C\)."
Step by step solution
01
Choose a Substitution
We'll use the substitution method to simplify the integral. Let \( u = x + 2 \). Then, differentiate both sides: \( du = dx \). This transforms the integral into terms of \( u \).
02
Rewrite the Function in Terms of u
Since \( u = x + 2 \), then \( x = u - 2 \). Substitute \( x = u - 2 \) and \( du = dx \) into the integral:\[\int \frac{(u-2) du}{u^4}\]
03
Simplify the Integral
Simplify the integral by breaking the fraction into two parts:\[\int \frac{(u-2)}{u^4} du = \int \left( \frac{u}{u^4} - \frac{2}{u^4} \right) du = \int (u^{-3} - 2u^{-4}) du\]
04
Perform the Integration
Integrate each term individually:\[\int u^{-3} du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}\]\[\int -2u^{-4} du = -2 \cdot \frac{u^{-3}}{-3} = \frac{2}{3u^3}\]Thus, the integrated function is:\[-\frac{1}{2u^2} + \frac{2}{3u^3} + C\]
05
Substitute Back to x
Substitute \( u = x + 2 \) back into the expression:\[-\frac{1}{2(x+2)^2} + \frac{2}{3(x+2)^3} + C\]This is the solution in terms of \( x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
When tackling integrals, several techniques can be used to find a solution. Understanding these techniques will help you simplify complex functions.
- Substitution Method: A technique where you substitute a part of the integral with a single variable, making it easier to integrate. This is often used when dealing with composite functions.
- Integration by Parts: Used when the integral is the product of two functions, allowing you to break it down into simpler parts.
- Partial Fraction Decomposition: Useful for rational functions, where you decompose into simpler fractions that can be integrated individually.
Substitution Method
The substitution method is a powerful integration technique, especially for simplifying functions involving composite terms. Here's how it works:
- Identify the Inner Function: In this method, look for a part of the integrand (the function inside the integral) that can act as a new variable. Here, we chose \( u = x + 2 \).
- Find Its Derivative: Differentiate your choice to find \( du \). For \( u = x + 2 \), we get \( du = dx \).
- Substitute and Simplify: Replace the variables in the integral with \( u \) and \( du \), transforming the integral into a simpler form to work with.
Definite Integral
Although our exercise dealt with an indefinite integral, understanding the concept of definite integrals is crucial. A definite integral has upper and lower limits and calculates the area under the curve of a function between these two bounds.
- Function with Bounds: Instead of an indefinite integral like \( \int f(x) \, dx \), a definite integral is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration.
- Net Result: The definite integral gives a specific numerical value representing the area, which can be positive, negative, or zero, depending on the curve and interval.
Indefinite Integral
An indefinite integral, unlike a definite integral, does not have set bounds and includes a constant of integration, \( C \). It represents a family of functions, all of which can differ by a constant.
- General Form: Indefinite integrals are expressed as \( \int f(x) \, dx \), without limits.
- Constant of Integration: The "+ C" represents that there are infinitely many antiderivatives. For example, the integral in our exercise results in \(-\frac{1}{2(x+2)^2} + \frac{2}{3(x+2)^3} + C\).
- Purpose: The indefinite integral is used to find antiderivatives and represented solutions to differential equations.
