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Chapter 8: Problem 35

Evaluate the integrals in Exercises \(31-52 .\) Some integrals do not require integration by parts. $$ \int \frac{\ln x}{x^{2}} d x $$

Short Answer

Expert verified
\(-\frac{\ln x + 1}{x} + C\)

Step by step solution

01

Identify the Integral Form

We need to solve the integral \( \int \frac{\ln x}{x^2} \, dx \). This problem seems to suit the integration by parts method, where we identify the functions to differentiate and integrate.
02

Choose u and dv

For integration by parts, choose \( u = \ln x \) as it simplifies upon differentiation, and \( dv = \frac{1}{x^2} \, dx \) because it's straightforward to integrate.
03

Differentiate u and Integrate dv

Differentiate \( u = \ln x \) to get \( du = \frac{1}{x} \, dx \). Integrate \( dv = \frac{1}{x^2} \, dx \) to obtain \( v = -\frac{1}{x} \).
04

Apply Integration by Parts Formula

Substitute into the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] which gives us \[ \int \frac{\ln x}{x^2} \, dx = \left( \ln x \right)\left(-\frac{1}{x}\right) - \int \left(-\frac{1}{x}\right) \left(\frac{1}{x} \right) \, dx \] simplifying to \(-\frac{\ln x}{x} + \int \frac{1}{x^2} \, dx \).
05

Simplify and Solve Remaining Integral

Compute \( \int \frac{1}{x^2} \, dx \) which is \( -\frac{1}{x} \). Substitute back to get \(-\frac{\ln x}{x} + (-\frac{1}{x}) \), simplifying to the final solution: \(-\frac{\ln x + 1}{x} + C \) 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.

Definite Integrals
Integrating a function over an interval involves finding the definite integral. Unlike indefinite integrals, definite integrals produce a number, representing the area under the curve of a function within given bounds. A definite integral can be expressed as: \[ \int_{a}^{b} f(x) \ dx \]where \(a\) and \(b\) are the lower and upper limits of integration, and \(f(x)\) is the function being integrated.
  • Limits \(a\) and \(b\) indicate the range over which you calculate the integral.
  • If the curve lies above the x-axis, the integral gives the area as a positive value.
  • If the curve is below the x-axis, the integral's value will be negative.
Understanding definite integrals is crucial because they are widely used in real-world applications, such as calculating displacement or determining area, making them an essential tool in calculus.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828.
  • The natural logarithm serves as the inverse function of exponential functions involving base \( e \).
  • When \( \ln x \) is differentiated, it simplifies into a derivative: \( \frac{d}{dx} \ln x = \frac{1}{x} \).
You can often encounter natural logarithms in functions that describe growth processes, like population growth or interest calculations. The simplicity of its derivative is why it is strategically chosen as \( u \) in integration by parts, as its differentiation helps in simplifying the integration process.
Antiderivatives
Antiderivatives, or indefinite integrals, are the reverse of differentiation. They allow us to find a function whose derivative is the original function. The general form of an antiderivative is:\[ \int f(x) \ dx = F(x) + C \]
  • \(F(x)\) represents the antiderivative of \(f(x)\), and \(C\) is the constant of integration.
  • Finding an antiderivative necessitates recognizing an existing pattern or function structure that is easy to reverse engineer.
  • The constant \(C\) is crucial because many functions can have derivatives that appear identical. Thus, \(C\) accounts for the family of solutions.
Understanding antiderivatives and applying them in problems like the integration by parts makes calculus not just a tool for solving problems but also gives insight into the behavior of functions and how they change.

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Most popular questions from this chapter

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{\ln 2} x^{-2} e^{-1 / x} d x$$

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