Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 1044

\(\pi \int_{-\pi}\left[\\{2 x(1+\sin x)\\} /\left(1+\cos ^{2} x\right)\right] d x\) is equal to....... (a) 0 (b) (c) \(\left(\pi^{2} / 2\right)\) (d) \(\pi^{2}\)

Short Answer

Expert verified
(a) 0

Step by step solution

01

Simplifying the integrand

Let's first rewrite the integral: \(\pi\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{1+\cos^2 x}dx\) Our goal is to simplify it to make the integration process easier. Notice that \(1+\cos^2x\) can be written as \(\sin^2x + \cos^2x\) (since \(\sin^2x + \cos^2x = 1\)). Now, we can rewrite the integral as: \(\pi\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{\sin^2 x + \cos^2 x}dx\)
02

Observing the symmetry of the integrand

We will now observe the symmetry of the integrand \(\frac{2x(1+\sin x)}{\sin^2 x + \cos^2 x}\) The given integral is an even function with respect to x multiplied by an odd function of x (\(2x(1+\sin x)\)) and the denominator is always even. Now, the integrand becomes an odd function overall since: \(odd * even = odd\) Since the given function is odd, we may say that: \(\int_{-a}^{a}f(x) dx = 0\), for any odd function Hence, the integral evaluates to \(0\). So, \(\boxed{\text{(a) 0}}\) is the correct answer.

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

The value of integral \(\pi \int_{0}[\\{\sin (2 n+1)(x / 2)\\} /\\{\sin (x / 2)\\}] d x\) is...... (a) 0 (b) \((\pi / 2)\) (c) \(\pi\) (d) \(2 \pi\)

The area of the region bounded by the lines \(y=m x, x=1\), \(\mathrm{x}=2\) and \(\mathrm{x}\) -axis is \(6 \mathrm{Sq}\). unit then \(\mathrm{m}\) is....... (a) 1 (b) 2 (c) 3 (d) 4

\(\pi \int_{0}[(\sin 100 \mathrm{x}) /(\sin \mathrm{x})] \mathrm{d} \mathrm{x}\) is equal to \(\ldots \ldots\) (a) 0 (b) \(\pi\) (c) \((\pi / 2)\) (d) \(2 \pi\)

If \(\mathrm{f}\) is an even function and \({ }^{2} \int_{0} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\mathrm{K}\) then \({ }^{1} \int_{-1}\left[\left(x^{2}-1\right) / x^{2}\right] f[x+(1 / x)] d x\) is equal to...... (a) 0 (b) \(2 \mathrm{~K}\) (c) \(\mathrm{K}\) (d) \(4 \mathrm{~K}\)

The area bounded by the curves \(x^{2}=y\) and \(2 x+y-8=0\) and \(y\) -axis in the second quadrant is...... (a) 9 Sq. unit (b) 18 Sq. unit (c) \((80 / 3)\) Sq. unit (d) 36 Sq. unit
See all solutions

Recommended explanations on English Textbooks

Summary Text

Read Explanation

Multiple Choice Questions

Read Explanation

Email

Read Explanation

Pragmatics

Read Explanation

English Language Study

Read Explanation

Semiotics

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