Problem 82

Question

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

Step-by-Step Solution

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Answer

The indefinite integral is $ 2x + 5 \ln |x| + C $.

1Step 1: 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 \[\int (2 + \frac{5}{x}) \, dx\].

2Step 2: 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 $.

3Step 3: 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.

Key Concepts

Fraction DecompositionTerm-by-Term IntegrationConstant of Integration

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.

Problem 81

Problem 82

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Problem 83

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