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Chapter 3: Problem 248

Use a table of integrals to evaluate the following integrals. $$ \int \frac{x}{x+1} d x $$

Short Answer

Expert verified
The integral is \( x - \ln |x+1| + C \).

Step by step solution

01

Identify Form Matching in Integral Table

We need to recognize a form from the integral table that matches our integral \( \int \frac{x}{x+1} dx \). This can be seen as a combination of integrals involving logarithmic or algebraic simplifications.
02

Simplify the Integral

Rewrite the integrand: \( \frac{x}{x+1} = 1 - \frac{1}{x+1} \). This allows us to express \( \int \frac{x}{x+1} dx \) as \( \int 1 \, dx - \int \frac{1}{x+1} \, dx \).
03

Integrate Each Term Separately

Now, split the integral into two parts: \( \int 1 \, dx = x \) and \( \int \frac{1}{x+1} \, dx = \ln |x+1| \).
04

Combine Results with Constant of Integration

Combine the integrated results: \( \int \frac{x}{x+1} \, dx = x - \ln |x+1| + C \), 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.

Integral Tables
Integral tables are like cheat sheets for calculus. They provide lists of common integrals and their results. This resource can save time because you don't always need to work out the integral from scratch. Instead, you find a matching form in the table and use the solution provided.

To use an integral table, you must first identify the standard form of your integral. This involves recognizing patterns and may sometimes require a little rewriting of your expression. Once you find a match, you simply apply the related formula.
  • Focus on the given integral's form and pattern.
  • Search for similar expressions in the table.
  • Apply the matching integral solution to your problem.
Using integral tables speeds up the integration process, making it easier and less error-prone.
Logarithmic Integration
Logarithmic integration is a technique used when dealing with integrals that involve logarithms. It comes in handy, especially when you have fractional forms where the integrand includes expressions like \( \frac{1}{x} \) or similar forms.

In the given exercise, after simplifying the original integral by rewriting it, we found a logarithmic form \( \int \frac{1}{x+1} \, dx \). This integral is an example where logarithmic integration becomes useful. The result of this specific integral is \( \ln |x+1| \), which is a common outcome.
  • Recognize forms that lead to logarithmic integration.
  • Remember the basic result: \( \int \frac{1}{x} \, dx = \ln |x| + C \).
  • Be careful with absolute values to ensure the solution is accurate for the domain.
Logarithmic integration simplifies the process of finding integrals involving rational functions.
Algebraic Simplification
Algebraic simplification involves rewriting expressions to make them easier to work with. It's a crucial step in integration. Before integrating, simplifying the integral can reveal a more straightforward path to the solution.

In this exercise, we reworked the integrand \( \frac{x}{x+1} \) by expressing it as \( 1 - \frac{1}{x+1} \). This transformation made it possible to break down the integral into simpler, more recognizable parts.
  • Identify parts of the expression that can be simplified.
  • Reorganize terms to reveal standard forms.
  • Check your work by substituting back to ensure equivalence.
Algebraic simplification is a powerful tool that often simplifies complex problems into manageable tasks.

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Most popular questions from this chapter

Find the volume of the solid generated by revolving about the \(y\) -axis the region under the curve \(y=6 e^{-2 x}\) in the first quadrant.

Evaluate the integrals. If the integral diverges, answer "diverges." $$ \int_{1}^{\infty} \frac{d x}{x^{e}} $$

Evaluate the integrals. If the integral diverges, answer "diverges." $$ \int_{0}^{e} \ln (x) d x $$

Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval. $$ \int_{6}^{24} \frac{d t}{t \sqrt{t^{2}-36}} $$

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge. $$ \int_{-1}^{2} \frac{d x}{x^{3}} $$
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