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Chapter 4: Problem 21

Determine these indefinite integrals. $$\int\left(\frac{3}{x}+\frac{5}{x^{2}}\right) d x$$

Short Answer

Expert verified
\( \int \left( \frac{3}{x}+\frac{5}{x^{2}} \right) dx = 3 \ln|x| - \frac{5}{x} + C \)

Step by step solution

01

Break Down the Integral

Separate the given integral into two separate integrals: \[ \ \int \left( \frac{3}{x} + \frac{5}{x^{2}} \right) dx = \int \frac{3}{x} \, dx + \int \frac{5}{x^{2}} \, dx \]
02

Solve the First Integral

Identify the antiderivative of the first term, \( \frac{3}{x} \). This is a standard integral: \[ \int \frac{3}{x} \, dx = 3 \int \frac{1}{x} \, dx = 3 \ln|x| \]
03

Simplify the Second Term

Rewrite the second term, \( \frac{5}{x^{2}} \), as \( 5x^{-2} \): \[ \int \frac{5}{x^{2}} \, dx = \int 5x^{-2} \, dx \]
04

Solve the Second Integral

Find the antiderivative of \( 5x^{-2} \): \[ \int 5x^{-2} \, dx = 5 \int x^{-2} \, dx = 5 \left( \frac{x^{-1}}{-1} \right) = -5x^{-1} = -\frac{5}{x} \]
05

Combine the Results

Combine the antiderivatives from Step 2 and Step 4, and don't forget to add the constant of integration, \( C \): \[ 3 \ln|x| - \frac{5}{x} + C \]

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Key Concepts

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

antiderivative
The term 'antiderivative' refers to finding a function whose derivative is the given function. Essentially, it's the process of reversing differentiation. When we take the derivative of a function, we are finding its rate of change. Conversely, finding an antiderivative allows us to reconstruct the original function given its derivative.
For example, if we know that the derivative of some function is \(\frac{3}{x}\), an antiderivative for this function is \((3 \ln|x|)\), since differentiating \((3 \ln|x|)\) gives us \(\frac{3}{x}\).
This process is fundamental in integration; every integral represents an antiderivative.
integration techniques
Several techniques help us find antiderivatives effectively.
In the given exercise, we used:
  • **Breaking down the integral:** We separated the original integral into two simpler integrals.
  • **Rewriting terms:** For instance, converting \(\frac{5}{x^{2}}\) to \(5x^{-2}\) which is more straightforward to integrate.

By breaking complex integrals into smaller parts or transforming them, we can apply known rules and formulas to find the solution more easily.
logarithmic functions
Logarithmic functions are integral in various areas of calculus. The natural logarithm, denoted \(\ln|x|\), often appears in integration.
In our exercise, the term \(\frac{3}{x}\) integrates to \(3 \ln|x|\). This result follows from the fact that the antiderivative of \(\frac{1}{x}\) is \(\ln|x|\).
Logarithmic functions help solve integrals involving terms like \(\frac{1}{x}\) or similar expressions, providing a straightforward antiderivative.
power rule for integration
The power rule for integration is a fundamental technique for finding antiderivatives. It states that to integrate \({x^n}\), where \(n \eq -1\), we use:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} \]
In our example, we applied this rule to \(5x^{-2}\). By integrating it, we get:
\[ \int 5x^{-2} \, dx = 5 \int x^{-2} \, dx = 5 \left( \frac{x^{-1}}{-1} \right) = -\frac{5}{x} \]
The power rule simplifies the process of integrating polynomial expressions or any term of the form \(x^n\).

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