Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 1042

\((\pi / 4) \int_{0} \log (\cot 2 x)^{\sin 4 x} d x\) is equal to \(\ldots \ldots\) (a) 0 (b) \((\pi / 4)\) (c) \((\pi / 8)\) (d) \((\pi / 2)\)

Short Answer

Expert verified
The given integral, $(\pi / 4) \int_{0}^{\pi/4} \log (\cot 2 x)^{\sin 4 x} d x$, is equal to \(\boxed{0}\).

Step by step solution

01

Recognize the limits of integration

Notice that the integral limits are from 0 to \(\pi/4\). This fact will be helpful when working with trigonometric functions like cotangent and sine.
02

Simplify the integrand using properties of logarithms

We can simplify the integrand using logarithm properties. Recall that \(\log(a^b) = b \log(a)\). Therefore, the integrand can be rewritten as: \[ \sin(4x) \log(\cot(2x)). \] Now the integral becomes: \[ (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx \]
03

Use substitution to deal with the cotangent function

Let \(u = \cot(2x)\). To find the corresponding differential, we will differentiate \(u\) with respect to \(x\): \[ \frac{du}{dx} = -2 \csc^2(2x). \] Now we can solve for \(dx\): \[ dx = -\frac{1}{2} \csc^2(2x) du. \] We also need to change the limits of integration. When \(x = 0\), we have \(u = \cot(0) = \infty\). When \(x = \pi/4\), we have \(u = \cot(\pi/2) = 0\). Now we can rewrite the integral in terms of \(u\): \[ (\pi / 4) \int_{\infty}^{0} -\frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \]
04

Recognize the symmetry of the integrand

Notice that the sine function is an odd function, meaning \(\sin(-x) = -\sin(x)\). Cotangent and cosecant squared are even functions, meaning \(\cot(-x) = \cot(x)\) and \(\csc^2(-x) = \csc^2(x)\). Therefore, we can rewrite the integral as follows: \[ (\pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(-4x) \log(u) \csc^2(-2x) du. \] Now, we can use the fact that the sine function is odd and that cotangent and cosecant squared are even functions to rewrite the integral again: \[ -(\pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \]
05

Recognize that the integrand is zero

Since the original integral and the rewritten integral in steps 4 and 5 have equal values, we can write the following equation: \[ (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx = -( \pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \] Adding these two integrals, we get: \[ 2 (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx = 0. \] Finally, dividing by 2 and multiplying by \(\pi/4\), we get: \[ (\pi / 4) \int_{0}^{\pi/4} \log (\cot 2 x)^{\sin 4 x} d x = 0. \] Thus, the correct answer is (a) 0.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \(\mathrm{f}\) is an odd function the value of integral \(\mathrm{e} \int_{(1 / \mathrm{e})}(1 / \mathrm{x}) \mathrm{f}[\mathrm{x}-(1 / \mathrm{x})] \mathrm{d} \mathrm{x}\) is equal to...... (a) \(\mathrm{e}\) (b) \(\left[\left(e^{2}+1\right) / e\right]\) (c) \(\left[\left(e^{2}-1\right) / 2 e\right]\) (d) 0

The value of \((\pi / 4)\\}_{0}\left[\left(8 \tan ^{2} x+8 \tan x+8\right)\right.\) \(\left./\left(\tan ^{2} x+2 \tan x+1\right)\right] d x\) is \(\ldots \ldots\) (a) 0 (b) \(\pi\) (c) \(\pi+2\) (d) \(\pi-2\)

\((\pi / 2) \int_{-(3 \pi / 2)}\left[(x+\pi)^{3}+\cos ^{2}(x+3 \pi)\right] \mathrm{d} x\) is equal to..... (a) \(\left(\pi^{3} / 8\right)\) (b) \((\pi / 2)\) (c) \((\pi / 4)-1\) (d) \((\pi / 4)+1\)

The value of the integral \({ }^{1} \int_{0}\left(x^{5}+6 x^{4}+5 x^{3}+4 x^{2}+3 x+1\right) e^{x-1} d x\) is equal to...... (a) 5 (b) \(5 \mathrm{e}\) (c) \(5 \mathrm{e}^{2}\) (d) \(5 \mathrm{e}^{4}\)

\((\pi / 2) \int_{0} \sqrt{(\sec x+1) d x}\) is equal to \(\ldots \ldots\) (a) 0 (b) \((\pi / 4)\) (c) \((\pi / 2)\) (d) \(\pi\)
See all solutions

Recommended explanations on English Textbooks

Phonetics

Read Explanation

Argumentative Essay

Read Explanation

Graphology

Read Explanation

English Grammar Summary

Read Explanation

Text Comparison

Read Explanation

Creative Story

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.

Sign-up for free