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Chapter 7: Problem 2

Find the indefinite integral without using a table: \(\int \frac{1}{x(x-a)} d x\).

Short Answer

Expert verified
The integral \(\int \frac{1}{x(x-a)} dx\) is \(ln|\frac{x}{x-a}|\) when simplified.

Step by step solution

01

Perform Partial Fraction Decomposition

To start the problem, rewrite the integral in a more manageable form by using partial fraction decomposition. The integral \(\int \frac{1}{x(x-a)} dx\) can be written as \(\int \frac{A}{x} + \frac{B}{x-a} dx\), where A and B are constants that can be found by setting the sum of the fractions equal to the original fraction.
02

Solve for A and B

By setting \(\frac{1}{x(x-a)}\) equal to \(\frac{A}{x} + \frac{B}{x-a}\) and clearing the denominators, you get the equation 1=A(x-a)+Bx. By comparing coefficients, you can see that B=-A. Substituting the value of A=1 (which was obtained by setting x=0), B is also found to be -1.
03

Substitute A and B into the Integral

Substitute the values of A and B that were found in Step 2 into the integral. This gives \(\int \frac{1}{x} - \frac{1}{x-a} dx\).
04

Integrate Each Term

Now, it is straightforward to integrate each term of the integral. The integral of \(\frac{1}{x} dx\) is \(ln|x|\), and the integral of \(-\frac{1}{x-a} dx\) is \(- ln|x-a|\). This gives \(ln|x| - ln|x-a|\).
05

Simplify by Using Logarithmic Properties

Using the logarithmic property \(ln(a) - ln(b) = ln(\frac{a}{b})\), the final answer is simplified to \(ln|\frac{x}{x-a}|\).

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Key Concepts

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

Partial Fraction Decomposition
Partial fraction decomposition is a technique used to integrate rational functions more easily. It takes a complex fraction and breaks it down into simpler fractions. In the exercise, we want to decompose \( \frac{1}{x(x-a)} \).
This is accomplished by expressing the fraction as a sum of two simpler fractions: \( \frac{A}{x} + \frac{B}{x-a} \).

Here鈥檚 how it works:
  • First, rewrite the original fraction under consideration as a sum of partial fractions. In our case, it is set up as \( \frac{A}{x} + \frac{B}{x-a} \).
  • Then, find A and B by multiplying both sides by the denominator, \( x(x-a) \), to get rid of the fractions. This yields the equation \( 1 = A(x-a) + Bx \).
  • After deciding on strategic values for x, like \( x = 0 \) and \( x = a \), you can solve for A and B.
This method makes integration more approachable by turning it into a task of integrating simpler fractions one by one.
Integration Techniques
Integration techniques are fundamental methods used to solve integrals that are not straightforward. Once the partial fraction decomposition is complete, the integral can be approached in a simpler format. In this case, we are left with \( \int \frac{1}{x} - \frac{1}{x-a} \, dx \). There are some key techniques for integrating such functions:

  • Direct Integration: Once resolved into simpler parts, direct integration is often the easiest method. Each simple fraction is integrated independently.
  • The integral of \( \frac{1}{x} \, dx \) results in \( \ln|x| \), a common result because the derivative of \( \ln|x| \) is \( \frac{1}{x} \).
  • With \( \frac{1}{x-a} \, dx \), you integrate it as if it were \( \frac{1}{u} \, du \), resulting in \( \ln|x-a| \).
Breaking down complex integrals into simple parts allows application of these basic techniques effectively, facilitating smoother computation.
Logarithmic Properties
Logarithmic properties can be powerful tools for simplifying expressions after integration. Once we integrate and obtain \( \ln|x| - \ln|x-a| \), we can simplify it further using properties of logarithms.

Some key properties are:
  • One important property is \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This simplifies the expression from two logarithmic terms into a single term.
  • This property helps to condense and make the solution more elegant and interpretable.
  • In the given problem, applying this property gives us \( \ln\left|\frac{x}{x-a}\right| \) as the final result.
Using logarithmic properties reduces complex-looking mathematical results, making them easier to interpret and apply in future applications.

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Most popular questions from this chapter

Evaluate the definite integral: \(\int_{2}^{4} \frac{1}{x \ln (x)} \mathrm{d} x\).

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Evaluate the definite integral: \(\int_{0}^{\pi / 2} \sin (x) \cos (x) \mathrm{d} x\).

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