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Chapter 4: Problem 19

Evaluate the indicated integral. $$\int \frac{4}{x(\ln x+1)^{2}} d x$$

Short Answer

Expert verified
The indefinite integral of \( \frac{4}{x(\ln x+1)^{2}} \) dx is \( -4(\ln{x} + 1)^{-1} + C \).

Step by step solution

01

Use Substitution

We use the substitution \( u = \ln{x} + 1 \). This leads to du = \( \frac{1}{x} \) dx. We can solve for dx, giving \( dx = x \) du. In our integral, \( x \) and dx will cancel, simplifying the integrand. The integral now becomes \( \int \frac{4}{u^{2}} \) du.
02

Evaluate Integral

The given integral is a power rule problem. Using power rule for integrals, we integrate \( \int u^{-2} \) du. The integral of \( u^{-2} \) with respect to \( u \) is \( -u^{-1} + C \). When we multiply by 4, the integral becomes \( -4u^{-1} + C \).
03

Back Substitute u

Now, we substitute \( u = \ln{x} + 1 \) back into the answer. This gives the final answer to be \( -4(\ln{x} + 1)^{-1} + C \).

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Key Concepts

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

Integration Techniques
Integration is a fundamental concept in calculus that helps us find areas under curves and solve real-world problems involving accumulation. Several techniques can efficiently solve different types of integrals. Let's explore some commonly used ones:
  • The Substitution Method simplifies integrals by changing variables.
  • The Partial Fraction Decomposition breaks down complex rational expressions.
  • The Integration by Parts is useful when the integrand is a product of functions.
  • Trigonometric Integration applies when integrals involve trigonometric functions.
Each method can be the key to solving an integral. The choice of technique depends heavily on the form of the integrand. Understanding and practicing these techniques will enhance your problem-solving skills in calculus.
Substitution Method
The substitution method, also known as u-substitution, is a powerful technique that simplifies the process of integration by substituting a part of the integrand with a new variable.This method transforms a complicated integral into a more manageable one.
Here's how it generally works:
  • First, select a part of the integrand to substitute with a new variable, say, \( u \).
  • Then, compute the differential, \( du \), in terms of the original variable.
  • Replace the parts of the integrand and the differential in terms of \( u \), simplifying the integral.
  • Finally, integrate with respect to \( u \) and substitute back the original variable using the relationship defined by \( u \).
In our original exercise, we used \( u = \ln{x} + 1 \). By changing the variable, the integral simplifies significantly, making it easier to solve with the power rule for integrals.
Power Rule for Integrals
The power rule for integrals is a basic yet highly useful technique in calculus, applicable when integrating power functions. It's a direct extension of the power rule for derivatives.Here's the rule:
  • For a function of the form \( x^n \), the integral is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
  • The constant \( C \) represents the constant of integration, accounting for indefinite integrals that could differ by a constant.
However, if \( n = -1 \), the integral becomes the natural logarithm, \( \int x^{-1} \, dx = \ln|x| + C \).

In our step-by-step solution, the function was \( u^{-2} \). Following the power rule, integrating \( u^{-2} \) gives \( -u^{-1} + C \), which becomes the term \( -4u^{-1} + C \) after multiplying by 4. Such applications emphasize the importance of this rule when dealing with polynomial-like functions in integrals.

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Most popular questions from this chapter

Evaluate the integral. $$\int_{0}^{1} \tan x d x$$

Identify the integrals to which the Fundamental Theorem of Calculus applies; the other integrals are called improper integrals. (a) \(\int_{0}^{1} \frac{1}{\sqrt{x+2}} d x \quad\) (b) \(\int_{0}^{2} \frac{1}{(x-3)^{2}} d x \quad\) (c) \(\int_{0}^{2} \sec x d x\)

The velocity of an object at various times is given. Use the data to estimate the distance traveled. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 26 & 30 & 28 & 30 & 28 & 32 & 30 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 14 & 16 & 18 & 20 & 22 & 24 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 33 & 31 & 28 & 30 & 32 & 32 \\ \hline \end{array}$$

For \(a>0,\) show that \(\int_{a}^{1} \frac{1}{x^{2}+1} d x=\int_{1}^{1 / a} \frac{1}{x^{2}+1} d x .\) Use this equality to derive an identity involving tan \(^{-1} \bar{x}\)

The data come from a pneumotachograph, which measures air flow through the throat (in liters per second). The integral of the air flow equals the volume of air exhaled. Estimate this volume. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 \\ \hline f(t)(1 / s) & 0 & 0.2 & 0.4 & 1.0 & 1.6 & 2.0 & 2.2 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 \\ \hline f(t)(1 / \mathrm{s}) & 2.0 & 1.6 & 1.2 & 0.6 & 0.2 & 0 \\ \hline \end{array}$$
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