Problem 81

Question

Compute the indefinite integrals. $$ \int \frac{2 x-1}{3 x} d x $$

Step-by-Step Solution

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Answer

The indefinite integral is $ \frac{2}{3}x - \frac{1}{3}\ln|x| + C $.

1Step 1: Simplify the Fraction

Start by simplifying the fraction inside the integral to make integration easier. Rewrite the integrand as two separate fractions: $ \frac{2x}{3x} - \frac{1}{3x} $. This simplifies to $ \frac{2}{3} - \frac{1}{3x} $. Then, the integral becomes $ \int \left( \frac{2}{3} - \frac{1}{3x} \right) dx $.

2Step 2: Integrate Each Term Separately

Proceed by integrating each term independently. First, integrate $ \frac{2}{3} $, which gives $ \frac{2}{3}x $. Next, integrate $ -\frac{1}{3x} $, which involves a natural logarithm: $ -\frac{1}{3} \ln|x| $.

3Step 3: Combine the Results and Add the Constant of Integration

Combine the results from Step 2. The complete integral is $ \frac{2}{3}x - \frac{1}{3}\ln|x| + C $, where $ C $ is the constant of integration.

Key Concepts

Simplifying FractionsNatural LogarithmConstant of Integration

Simplifying Fractions

Simplifying fractions is a vital skill in calculus and algebra since it aids in clarity and ease of further computation. When faced with an expression, breaking down a fraction into simpler components can provide a clearer path to integration or differentiation.

To simplify the fraction $ \frac{2x - 1}{3x} $, we separate it into two distinct terms, leading to $ \frac{2x}{3x} - \frac{1}{3x} $.
This separation can help clarify each term, transforming it to $ \frac{2}{3} - \frac{1}{3x} $.

This step makes the upcoming integration more straightforward, avoiding unnecessary complications in calculation. It's always beneficial to check if the denominator can help separate the terms and, if possible, simplify them directly, as was the case here.

Natural Logarithm

The natural logarithm, often denoted as $ \ln $, is a special logarithm that uses the constant $ e $ (approximately 2.718) as its base. In integration, it frequently appears when dealing with terms of the form $ \frac{1}{x} $.

In our integral, we encountered $ -\frac{1}{3x} $, which transforms into $ -\frac{1}{3} \ln|x| $ upon integration.
This transformation occurs because the derivative of $ \ln|x| $ is $ \frac{1}{x} $.
Remember, the absolute value in $ \ln|x| $ ensures that the logarithm remains defined for all non-zero values of $ x $.

Using the natural logarithm is crucial in this step because it is the elegant solution for integrals involving $ \frac{1}{x} $. It's versatile and can handle both positive and negative values of $ x $ due to the absolute value.

Constant of Integration

The constant of integration is an essential concept when dealing with indefinite integrals. It represents an infinite set of possible functions that differ only by their constant term.

Whenever you integrate a function indefinitely, the result may differ by a constant, because the derivative of a constant is zero.
In our solution, the expression became $ \frac{2}{3}x - \frac{1}{3}\ln|x| + C $, where $ C $ is this arbitrary constant.
Including $ C $ ensures that you account for any vertical shifts of the function's graph that would result in an identical slope (derivative).

It's important to always include the constant of integration when finding indefinite integrals. Failing to do so can lead to incomplete solutions, as you would miss out on the entire family of curves that belong to the integral's solution.

Problem 80

Problem 81

Other exercises in this chapter

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Compute the indefinite integrals. $$ \int \frac{1}{x-3} d x $$

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Problem 82

Compute the indefinite integrals. $$ \int \frac{2 x+5}{x} d x $$

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