/*! 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 11 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 11

Evaluate the following integrals. $$\int \frac{\ln 2 x}{x} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{\ln 2x}{x}dx.$$ Answer: $$\int \frac{\ln 2x}{x}dx = \frac{1}{2}(\ln 2x)(\ln |x|) + C$$

Step by step solution

01

Choose u and dv

Let \(u = \ln 2x\) and \(dv = \frac{1}{x} dx\)
02

Calculate du and v

Now we need to calculate du and v. For du: \(du = \frac{d}{dx}(\ln 2x)dx = \frac{1}{x}dx\) For v: \(v = \int dv = \int \frac{1}{x} dx = \ln |x|\)
03

Apply integration by parts

Using the integration by parts formula: $$\int \frac{\ln 2 x}{x} dx = \int u \, dv = uv - \int v \, du$$ So we have, $$\int \frac{\ln 2 x}{x} dx = (\ln 2x)(\ln |x|) - \int (\ln |x|) \frac{1}{x}dx$$
04

Evaluate the remaining integral

The remaining integral is the same form as the initial integral, so we can just write it in terms of the initial integral: $$\int \frac{\ln 2 x}{x} d x =(\ln 2x)(\ln |x|) - \int \frac{\ln 2 x}{x} d x$$
05

Solve for the initial integral

Now, we can solve for the initial integral by moving it to the other side: $$2\int \frac{\ln 2 x}{x} d x =(\ln 2x)(\ln |x|)$$ Divide both sides by 2: $$\int \frac{\ln 2 x}{x} d x =\frac{1}{2}(\ln 2x)(\ln |x|) + C$$ So, the final answer is: $$\int \frac{\ln 2 x}{x} d x =\frac{1}{2}(\ln 2x)(\ln |x|) + C$$ where C is the constant of integration.

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

Graph the function \(f(x)=\frac{1}{x \sqrt{x^{2}-36}}\) on its domain. Then find the area of the region \(R_{1}\) bounded by the curve and the \(x\) -axis on \([-12,-12 / \sqrt{3}]\) and the area of the region \(R_{2}\) bounded by the curve and the \(x\) -axis on \([12 / \sqrt{3}, 12] .\) Be sure your results are consistent with the graph.

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \sin 3 x \sin 2 x d x$$

On the interval \([0,2],\) the graphs of \(f(x)=x^{2} / 3\) and \(g(x)=x^{2}\left(9-x^{2}\right)^{-1 / 2}\) have similar shapes. a. Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,2] b. Find the area of the region bounded by the graph of \(g\) and the \(x\) -axis on the interval [0,2] c. Which region has the greater area?

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{x}{(x-1)\left(x^{2}+2 x+2\right)^{2}} d x$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.