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Chapter 11: Problem 16

Evaluate the indefinite integral. $$ \int \frac{d x}{16 x^{4}-8 x^{2}+1} $$

Short Answer

Expert verified
The indefinite integral evaluates to

Step by step solution

01

- Identify Type of Integral

Observe that the integral involves a rational function with a polynomial in the denominator. Specifically, it is required to factor the quadratic in the denominator if possible.
02

- Factor the Denominator

Consider rewriting the denominator as a square. Recognize that the denominator can be factored using the form of a perfect square: 16x^4 - 8x^2 + 1 = (4x^2 - 1)^2. So, the integral becomes: = = = =
03

- Substitute the Variable

Rewriting the integral in terms of u: : . Finally, integrate with respect to u: .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are mathematical expressions that consist of a ratio of two polynomials. For example, \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. The function in our exercise is an example of a rational function: \( \frac{1}{16 x^{4}-8 x^{2}+1} \). Rational functions can often be integrated by breaking them down into simpler parts, which often involves factorization and substitution.
Factoring Polynomials
Factoring polynomials is the process of breaking down a complex polynomial into simpler components that, when multiplied together, produce the original polynomial. In our exercise, we dealt with factorizing the denominator of the rational function: \(16x^4 - 8x^2 + 1\). This polynomial is not immediately recognizable, but with some rewriting, we can reveal that it is a perfect square: \( (4x^2 - 1)^2 \). Here's how the step works:
  • 1. Identify any patterns that suggest a perfect square.
  • 2. Confirm by expanding to check your work.
Recognizing this makes subsequent steps, such as substitution, much easier.
Substitution Method
The substitution method simplifies an integral by changing variables. It involves choosing a new variable to replace an existing part of the integrand. In our problem, after factoring, we substitute \(u\) as follows: \(u = 4x^2-1\). This rewrites the integral in terms of \(u\) and simplifies it significantly:
  • 1. Identify what part of the integrand makes sense to substitute.
  • 2. Express the integral in terms of the new variable.
  • 3. Integrate with respect to the new variable.
This approach transforms the integral into something more manageable, allowing us to find the solution efficiently.

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