/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 17

Evaluate the following integrals. $$\int_{1}^{e^{2}} \frac{\ln ^{2}\left(x^{2}\right)}{x} d x$$

Short Answer

Expert verified
The value of the integral $$\int_{1}^{e^{2}} \frac{\ln ^{2}\left(x^{2}\right)}{x} d x$$ is equal to $$\frac{32}{3}$$ after simplifying the expression, making a substitution, evaluating the new integral, replacing back the substitution, and finally, evaluating the integral within the given limits.

Step by step solution

01

Simplify the Integral

First, let's simplify the expression inside the integral using properties of logarithms. We know that $$\ln\left(x^{2}\right) = 2\ln x$$ So, $$\ln^{2}\left(x^{2}\right) = (2\ln x)^{2}=4(\ln x)^{2}$$ Now, we can rewrite the integral as: $$\int_{1}^{e^{2}} \frac{4(\ln x)^{2}}{x} d x$$
02

Make a Substitution

Let's make the substitution $$u = \ln x$$ Then, differentiating both sides, we get $$d u = \frac{d x}{x}$$ Now substitute these expressions into the integral and change the limits according to the new variable: $$\int_{1}^{e^{2}} \frac{4(\ln x)^{2}}{x} d x = \int_{0}^{2} 4u^{2} d u$$
03

Evaluate the New Integral

Now, we will perform the integration with respect to u: $$\int 4u^{2} d u = 4\int u^{2} d u = 4\frac{u^{3}}{3} + C$$
04

Replace Back the Substitution

Now, we will replace back the substitution u = ln(x): $$4\frac{u^{3}}{3} + C = 4\frac{(\ln x)^{3}}{3} + C$$
05

Evaluate the Integral within the Given Limits

Finally, we will evaluate the integral within the given limits: $$\int_{1}^{e^{2}} \frac{4(\ln x)^{2}}{x} d x = 4\frac{(\ln x)^{3}}{3}\Big|_{1}^{e^{2}}$$ Applying the limits, we get: $$4\frac{(\ln (e^{2}))^{3}}{3} - 4\frac{(\ln 1)^{3}}{3} = 4\frac{(2)^{3}}{3} - 0 = 4\frac{8}{3} = \boxed{\frac{32}{3}}$$ Hence, the value of the integral is: $$\int_{1}^{e^{2}} \frac{\ln ^{2}\left(x^{2}\right)}{x} d x=\frac{32}{3}$$

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!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

The region bounded by \(f(x)=(4-x)^{-1 / 3}\) and the \(x\) -axis on the interval [0,4) is revolved about the \(y\) -axis. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(x\) -axis on the interval (-1,1] is revolved about the line \(y=-1\).

Evaluate the following integrals or state that they diverge. $$\int_{0}^{9} \frac{d x}{(x-1)^{1 / 3}}$$

Compute \(\int_{0}^{1} \ln x d x\) using integration by parts. Then explain why \(-\int_{0}^{\infty} e^{-x} d x\) (an easier integral) gives the same result.

An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem \(B^{\prime}(t)=a B-m\) for \(t \geq 0,\) with \(B(0)=B_{0} .\) The constant \(a\) reflects the annual interest rate, \(m\) is the annual rate of withdrawal, and \(B_{0}\) is the initial balance in the account. a. Solve the initial value problem with a=0.05, m= 1000 dollar \(/\mathrm{yr}\), and \(B_{0}\)= 15,000 dollar. Does the balance in the account increase or decrease? b. If \(a=0.05\) and \(B_{0}\)= 50,000 dollar, what is the annual withdrawal rate \(m\) that ensures a constant balance in the account? What is the constant balance?

Given a Midpoint Rule approximation \(M(n)\) and a Trapezoid Rule approximation \(T(n)\) for a continuous function on \([a, b]\) with \(n\) subintervals, show that \(T(2 n)=(T(n)+M(n)) / 2\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.