Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 949

\(\int\left[\left(x^{2} d x\right) /(x \sin x+\cos x)^{2}\right]=\ldots+c\) (a) \([(\sin x-x \cos x) /(x \sin x+\cos x)]\) (b) \([(\sin x+x \cos x) /(x \sin x+\cos x)]\) (c) \([(x \sin x-\cos x) /(x \sin x+\cos x)]\) (d) \([(x \sin x+\cos x) /(x \sin x-\cos x)]\)

Short Answer

Expert verified
The short answer is: \(\int\left[\left(x^{2} d x\right) /(x \sin x+\cos x)^{2}\right]=\frac{(x \sin x-\cos x)}{(x\sin x+\cos x)x^2} + C\).

Step by step solution

01

Integration by substitution

Let's perform integration by substitution. We'll substitute the denominator of the integrand with a new variable. Let \[u = x\sin x + \cos x\] Then, differentiate u with respect to x to get its differential: \[du/dx = (\sin x + x\cos x) -\sin x\] Which simplifies to: \[du/dx = x\cos x\] Now, multiply both sides by dx: \[du = x\cos x dx \implies (1/x^2) du = dx\]
02

Rewrite the integral in terms of u

Now, substitute \(u\) and the differential \(du\) back into the integral, so we get: \[\int\left[\left(\frac{1}{x^2} du\right) /(u^2)\right]\] This simplifies to: \[\int\frac{1}{u^2x^2}du\]
03

Integration of the transformed integral

Now, integrate the transformed integral with respect to u: \[\int\frac{1}{u^2x^2}du = -\frac{1}{ux^2} + C\]
04

Back substitution of x values

To express our antiderivative in terms of x, we perform the back substitution of the relationship for u in terms of x: \[u = x\sin x + \cos x\] So, the antiderivative is now: \[-\frac{1}{(x\sin x + \cos x)x^2} + C\]
05

Comparing the result and determining the correct option

Now, let's compare our antiderivative with the given options: (a) \([(\sin x-x \cos x) /(x \sin x+\cos x)]\) (b) \([(\sin x+x \cos x) /(x \sin x+\cos x)]\) (c) \([(x \sin x-\cos x) /(x \sin x+\cos x)]\) (d) \([(x \sin x+\cos x) /(x \sin x-\cos x)]\) Our antiderivative is \(-\frac{1}{(x\sin x + \cos x)x^2} + C\). On rearranging, it matches with option (c) after bringing both x and the constant term under the same denominator: \[\frac{(x \sin x-\cos x)}{(x\sin x+\cos x)x^2} + C\] So, the correct antiderivative for the given integral is: \[\int\left[\left(x^{2} d x\right) /(x \sin x+\cos x)^{2}\right]=\frac{(x \sin x-\cos x)}{(x\sin x+\cos x)x^2} + C\]

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.

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 \(\int\left[\mathrm{dx} /\left(\sin ^{6} \mathrm{x}+\cos ^{6} \mathrm{x}\right)\right]=\mathrm{K} \tan ^{-1}[(\tan 2 \mathrm{x}) / 2]+\mathrm{c}\) then \(\mathrm{K}=\) (a) \((\overline{1 / 2)}\) (b) \(-1\) (c) 1 (d) \(-(1 / 2)\)

\(\int\left[\left(\cos ^{8} x-\sin ^{8} x\right) /\left(1-2 \sin ^{2} x \cos ^{2} x\right)\right] d x=\underline{ }+c\) (a) \(-[(\cos 2 \mathrm{x}) / 2]\) (b) \(\overline{-[(\sin 2} x) / 2]\) (c) \([(\cos 2 \mathrm{x}) / 2]\) (d) \([(\sin 2 \mathrm{x}) / 2]\)

\(\int \operatorname{cosec}[\mathrm{x}-(\pi / 6)] \operatorname{cosec}[\mathrm{x}-(\pi / 3)] \mathrm{d} \mathrm{x}\) \(=\mathrm{k}[\log |\sin \\{\mathrm{x}-(\pi / 6)\\}|-\log |\sin \\{\mathrm{x}-(\pi / 3)\\}|]+\mathrm{c}\) then \(\mathrm{k}=\) (a) 2 (b) \(-2\) (c) \((\sqrt{3} / 2)\) (d) \((2 \overline{\sqrt{3})}\)

\(\int e^{2 x}(1+\tan x)^{2} d x=\) \(+c\) (a) \(\tan \mathrm{e}^{\mathrm{x}}\) (b) \(\tan \mathrm{x} \mathrm{e}^{2 \mathrm{x}}\) (c) \(\tan (\mathrm{x} / 2) \mathrm{e}^{\mathrm{x}}\) (d) \(\tan (\mathrm{x} / 2) \mathrm{e}^{-\mathrm{x}}\)

If \(\int\left[\left(2 \mathrm{e}^{\mathrm{x}}+3 \mathrm{e}^{-\mathrm{x}}\right) /\left(3 \mathrm{e}^{\mathrm{x}}+4 \mathrm{e}^{-\mathrm{x}}\right)\right] \mathrm{d} \mathrm{x}=\mathrm{Ax}+\mathrm{B} \log \left|3 \mathrm{e}^{2 \mathrm{x}}+4\right|+\mathrm{c}\) then \(\mathrm{A}+\mathrm{B}=\) (a) \((11 / 24)\) (b) \((13 / 24)\) (c) \((15 / 24)\) (d) \((17 / 24)\)
See all solutions

Recommended explanations on English Textbooks

Rhetorical Analysis Essay

Read Explanation

Research and Composition

Read Explanation

Discourse

Read Explanation

Language Analysis

Read Explanation

Rhetoric

Read Explanation

Text Comparison

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