Problem 97 Evaluate. Assume $u>0$ when... [FREE SOLUTION]

91影视

Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 97

Evaluate. Assume $u>0$ when ln u appears. $$\int \frac{(\ln x)^{n}}{x} d x, \quad n \neq-1$$

Short Answer

Expert verified

\int \frac{(\ln x)^n}{x} dx = \frac{(\ln x)^{n+1}}{n+1} + C.

Step by step solution

Identify the integral and make a substitution

Given the integral $\int \frac{(\ln x)^{n}}{x} dx$. Make the substitution $u = \ln x$. Then $du = \frac{1}{x} dx$. This simplifies the integral to $\int u^n du$

Integrate using the power rule

The integral $\int u^n du$ can be evaluated using the power rule of integration, which states $\int u^n du = \frac{u^{n+1}}{n+1} + C$, where $C$ is the constant of integration and $n eq -1$.

Substitute back in terms of x

Recall the substitution $u = \ln x$. Substitute back to get $\frac{(\ln x)^{n+1}}{n+1} + C$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Substitution Method

The substitution method is a popular technique in calculus for simplifying integrals. It's also known as $u$-substitution. Here's how it works:

You choose a new variable $u$ to substitute part of the integrand.
The goal is to transform the original integral into a simpler form.
You also need to change the differential $dx$ to $du$.

In our example, we have the integral $\to \int \frac{(\ln x)^{n}}{x} dx$. By setting $u = \ln x$, we simplify the integrand. The differential change is $du = \frac{1}{x} dx$. This allows us to rewrite the integral in terms of $u$: $\to \int u^n du$.
This new integral is much easier to evaluate.

Power Rule of Integration

The power rule of integration is a fundamental rule for integrating functions of the form $u^n$, where $n eq -1$.
The rule can be stated as follows:
\[ \int u^n du = \frac{u^{n+1}}{n+1} + C \]
Here, $C$ is the constant of integration. Notice how the exponent $n$ in the integrand simply increases by one, and we divide by this new exponent.
In our example, after substituting $u = \ln x$ and $du = \frac{1}{x} dx$, we end up with \[ \int u^n du = \frac{u^{n+1}}{n+1} + C \]
This process significantly simplifies the evaluation of the integral.

Natural Logarithm

The natural logarithm, denoted as $\ln x$, is a logarithm with base $e$ (where $e \approx 2.71828$). It's a crucial function in mathematics and appears frequently in calculus.
Here are some key points about the natural logarithm:

It is the inverse of the exponential function. This means $\ln(e^x) = x$ and $e^{\ln x} = x$.
Its derivative is ${\frac{d}{dx} \ln x} = \frac{1}{x}$.
It's often used in integration and differentiation problems.

In our example, we use the log's properties to rewrite an integral into a simpler form. We replaced $\ln x$ with a new variable $u$. After integration, we substituted back $u = \ln x$, giving the final result $\frac{{(\ln x)}^{n+1}}{{n+1}} + C$.
This showcases the utility of natural logarithms in calculus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate. Assume \(u>0\) when ln u appears. $$\int \frac{(\ln x)^{n}}{x} d x, \quad n \neq-1$$

Short Answer

Step by step solution

Identify the integral and make a substitution

Integrate using the power rule

Substitute back in terms of x

Key Concepts

Substitution Method

Power Rule of Integration

Natural Logarithm

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Discrete Mathematics

Decision Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.