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Chapter 6: Problem 82

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

Short Answer

Expert verified
The indefinite integral is \( 2x + 5 \ln |x| + C \).

Step by step solution

01

Split the Fraction

Break down the given function into simpler terms by dividing each term of the numerator by the denominator. This gives us \[\int \left( \frac{2x}{x} + \frac{5}{x} \right) \, dx\] which simplifies to \[\int (2 + \frac{5}{x}) \, dx\].
02

Integrate Term by Term

Integrate each term of the function separately. The integral of a constant 2 is \[2x\]. The integral of \( \frac{5}{x} \) is \[5 \ln |x|\]. This uses the rule \( \int \frac{1}{x} \, dx = \ln |x| + C \).
03

Combine the Results

Combine the integrals of the terms to obtain the complete integral. Thus, the integral of the original expression is \[2x + 5 \ln |x| + C\], where \( C \) is the constant of integration.

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Key Concepts

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

Fraction Decomposition
In calculus, simplifying complex fractions is a common step when dealing with integrals. This process usually involves breaking down a complicated fraction into simpler, more manageable terms. For example, in the given exercise, the fraction \( \frac{2x+5}{x} \) is split. Here's how you can do it:
  • Separate each term of the numerator.
  • Divide these terms individually by the denominator.
In our case, dividing both \(2x\) and \(5\) by \(x\) results in terms \(2\) and \(\frac{5}{x}\). This process simplifies the integral before solving it, making everything more approachable. Fraction decomposition is particularly useful when dealing with polynomials or rational functions.
Term-by-Term Integration
Once you've decomposed your fraction, it's time to integrate each term separately. Integrating term-by-term lets you focus on simpler, individual functions rather than a complex whole:
  • First, integrate the constant \(2\). For any constant \(a\), the rule is \(\int a \, dx = ax\).
  • Next, handle the term \(\frac{5}{x}\). This follows the basic integral \(\int \frac{1}{x} \, dx = \ln |x|\).
Therefore, the integral of \(\frac{5}{x}\) becomes \(5 \ln |x|\). Term-by-term integration allows you to employ different integration techniques as needed, based on the specific term you are dealing with. It's a methodical approach to tackling more complex integrals in parts.
Constant of Integration
The constant of integration, represented as \(C\), is a pivotal part of indefinite integrals. It emerges because integration can reverse derivatives, but it doesn't capture the constant vertical shifts of a function. Here's why it's important:
  • When you calculate an indefinite integral, you're essentially finding a family of functions that could fit the original derivative.
  • The constant \(C\) represents all possible vertical shifts of the resulting function.
  • Every valid antiderivative of a function includes an arbitrary constant, due to this reason.
With the exercise integral \(2x + 5 \ln |x| + C\), \(C\) means that there isn't just one solution, but infinitely many solutions, each differing by that constant value. Without it, you would not represent the complete set of antiderivative solutions possible.

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