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Magazine Chapter 5: Problem 49
Evaluate the integrals. $$\int \frac{x}{\left(x^{2}-4\right)^{3}} d x$$
Short Answer
Expert verified
The integral is \(-\frac{1}{4(x^2-4)^2} + C\).
Step by step solution
01
Identify the Method of Integration
To solve this integral, notice that the function inside the integrand is a composition involving a power of a binomial, \( (x^2 - 4)^3 \). This is a suitable scenario for substitution. We'll use the substitution method.
02
Perform the Substitution
Let \( u = x^2 - 4 \). Then, the derivative \( du = 2x \, dx \) can be rearranged to \( x \, dx = \frac{1}{2} du \). Substitute these into the integral.
03
Substitute and Simplify
Replace \( x^2 - 4 \) with \( u \) and \( x \, dx \) with \( \frac{1}{2} du \). The integral becomes: \[ \int \frac{x}{(x^2 - 4)^3} \ dx = \int \frac{1}{2u^3} \, du = \frac{1}{2} \int u^{-3} \, du \]
04
Integrate with Respect to \( u \)
Integrate \( \frac{1}{2} \int u^{-3} \, du \). The antiderivative of \( u^{-3} \) is \( \frac{u^{-2}}{-2} \) or \( -\frac{1}{2}u^{-2} \). Thus, the integral becomes: \[ \frac{1}{2} \left( -\frac{1}{2u^2} \right) = -\frac{1}{4u^2} + C \] , where \( C \) is the constant of integration.
05
Back-Substitute \( u \)
Replace \( u \) back with \( x^2 - 4 \). The expression \( -\frac{1}{4u^2} \) becomes \( -\frac{1}{4(x^2 - 4)^2} \). Consequently, the integral is: \[ -\frac{1}{4(x^2 - 4)^2} + C \]
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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 tool in calculus, designed to simplify complex integrals, making them more manageable to evaluate. It involves replacing the integrand with a new variable, usually denoted as \( u \), to transform the original integral into an easier form. The key steps for employing substitution are as follows:
- Identify a Substitution: Look for a part of the integrand that, if replaced, will simplify the integral. In this case, we noticed the binomial \((x^2 - 4)^3\) and chose \( u = x^2 - 4 \).
- Calculate the Derivative: Compute the derivative of the chosen substitution, \( du = 2x \, dx \), and express \( x \, dx \) as a function of \( du \). Here, \( x \, dx = \frac{1}{2} du \).
- Replace and Simplify: Substitute \( u \) and \( du \) into the integral to transform it. Our initial integral \( \int \frac{x}{(x^2 - 4)^3} \, dx \) simplifies to \( \int \frac{1}{2u^3} \, du \).
Definite and Indefinite Integrals
When dealing with integrals, it is essential to distinguish between definite and indefinite integrals. Each type serves a unique purpose and provides different information.
- Indefinite integrals: These represent a family of functions and include the constant of integration \( C \). They express the general form of antiderivatives. For example, integrating \( \int u^{-3} \, du \) results in \( -\frac{1}{2}u^{-2} + C \).
- Definite integrals: These provide the actual value of an area under a curve between two bounds, \([a, b]\). It results in a real number and does not include \( C \) because limits are involved in the calculation. Although our example is an indefinite integral, the substitution method can also be used for definite integrals with respective limits exchanged for \( u \).
Calculus Problems
Calculus problems vary widely, but typically involve differentiation or integration to solve. Mastering these concepts is crucial for understanding change and area in mathematical contexts.
- Problem Identification: Begin by identifying the part of the calculus problem you are dealing with: differentiation (solving derivatives) or integration (solving integrals). Our focus is on integration here.
- Choosing Techniques: Depending on the structure of the function, choose the suitable method to integrate. This could be substitution, integration by parts, partial fractions, etc. For our function, substitution was most appropriate.
- Apply and Simplify: Apply the chosen technique, simplify, and solve. Integration often requires iterative refinement and step-by-step simplification as shown in our worked solution.
