Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 923

If \(\int \sin ^{3} x d x=A \cos ^{3} x+B \cos x+c\) then \(A-B=\) (a) \((4 / 3)\) (b) \(-(4 / 3)\) (c) \((1 / 3)\) (d) \(\overline{-(1 / 3)}\)

Short Answer

Expert verified
The correct answer is none of the given options (a), (b), (c), and (d). The value of A-B is \(\dfrac{7}{9}\).

Step by step solution

01

Integrate sin^3(x) using Reduction Formula

To find the integration of sin^3(x) with respect to x, we will use the reduction formula for sin^n(x): \( \int \sin ^{n} x d x=-\dfrac{1}{n} (\cos x)(\sin ^{n-1} x)+\dfrac{n-1}{n} \int\sin ^{n-2} x d x \) So for our specific case (n=3), we have: \( \int \sin ^{3} x d x = -\dfrac{1}{3} (\cos x)(\sin ^{2} x)+\dfrac{2}{3} \int\sin ^{1} x d x \) Now we need to replace sin^2(x) with its identity sin^2(x)=1-cos^2(x). So, the equation becomes: \( \int \sin ^{3} x d x = -\dfrac{1}{3} (\cos x)(1-\cos ^{2} x)+\dfrac{2}{3} \int\sin x d x \) Next, we will integrate the remaining terms.
02

Evaluate the Integration

We will integrate the expression obtained after substitution: \( \int \sin ^{3} x d x = -\dfrac{1}{3} \int (\cos x-\cos ^{3} x) d x+\dfrac{2}{3} \int\sin x d x \) Now integrate each term: \( \int \sin ^{3} x d x = -\dfrac{1}{3} (\sin x-\dfrac{1}{3}\sin x \cos^2x)+\dfrac{2}{3}(-\cos x) + C \) \( \int \sin ^{3} x d x = -\dfrac{1}{3}\sin x + \dfrac{1}{9}\sin x \cos^2x -\dfrac{2}{3}\cos x + C \) Now compare this with the given expression: \( \int \sin ^{3} x d x = A \cos ^{3} x+B \cos x + c \) By comparing the coefficients, we get: \( A = \dfrac{1}{9}, B = -\dfrac{2}{3} \) Now calculate the value of (A-B):
03

Calculate A-B

We are given that A = 1/9 and B= -(2/3). Thus, \(A - B = \dfrac{1}{9} - (-\dfrac{2}{3}) = \dfrac{1}{9} + \dfrac{2}{3}\) Now, we will find the LCM which is 9: \(A - B = \dfrac{1}{9} + \dfrac{6}{9} = \dfrac{1+6}{9} = \dfrac{7}{9}\) The correct answer is none of the given options (a), (b), (c), and (d).

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 \(\int\left(x^{30}+x^{20}+x^{10}\right)\left(2 x^{20}+3 x^{10}+6\right)^{(1 / 10)} d x\) \(=\mathrm{k}\left(2 \mathrm{x}^{30}+3 \mathrm{x}^{20}+6 \mathrm{x}^{10}\right)^{(11 / 10)}+\mathrm{c}\) then \(\mathrm{k}=\) (c) \((1 / 66)\) (a) \((1 / 60)\) (b) \(-(1 / 60)\) (c) \(-(1 / 66)\)

\(\int\left[\left(x^{2} d x\right) /\left\\{\left(x^{2}+2\right)\left(x^{2}+3\right)\right\\}\right]=\ldots\) (a) \(\sqrt{3} \tan ^{-1}(\mathrm{x} / \sqrt{3})+\sqrt{2} \tan ^{-1}(\mathrm{x} / \sqrt{2})\) (b) \(\sqrt{3} \tan ^{-1}(\mathrm{x} / \sqrt{3})-\sqrt{2} \tan ^{-1}(\mathrm{x} / \sqrt{2})\) (c) \(\tan ^{-1}(\mathrm{x} / \sqrt{3})+\sqrt{2} \tan ^{-1}(\mathrm{x} / \sqrt{2})\) (d) \(\tan ^{-1}(\mathrm{x} / \sqrt{3})-\sqrt{2} \tan ^{-1}(\mathrm{x} / \sqrt{2})\)

If \(\int[(\cos 9 x+\cos 6 x) /(2 \cos 5 x-1)] d x=k_{1} \sin 4 x+k_{2} \sin x+c\) then \(4 \mathrm{k}_{1}+\mathrm{k}_{2}=\) (a) 1 (b) 2 (c) 4 (d) 5

\(\int\left[\left(2^{\sqrt{x}}\right) /(\sqrt{x})\right] d x=\) (a) \(2^{\sqrt{x}} \log _{2} \mathrm{e}\) (b) \(2^{\sqrt{x}} \log _{e} 2\) (c) \(2^{(\sqrt{x}+1)} \log _{2} e\) (d) \(2^{\sqrt{(x+1)}} \log _{\mathrm{e}} 2\)

\(\int\left[\left(\mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}\right) /\left\\{\left(\mathrm{e}^{\mathrm{x}}+2012\right)\left(\mathrm{e}^{\mathrm{x}}+2013\right)\right\\}\right]=\ldots\) (a) \(\log \left[\left(\mathrm{e}^{\mathrm{x}}+2012\right) /\left(\mathrm{e}^{\mathrm{x}}+2013\right)\right]\) (b) \(\log \left[\left(e^{x}+2013\right) /\left(e^{x}+2012\right)\right]\) (c) \(\left[\left(e^{x}+2012\right) /\left(e^{x}+2013\right)\right]\) (d) \(\left[\left(e^{x}+2013\right) /\left(e^{x}+2012\right)\right]\)
See all solutions

Recommended explanations on English Textbooks

Essay Plan

Read Explanation

English Language Study

Read Explanation

International English

Read Explanation

English Grammar

Read Explanation

Syntax

Read Explanation

5 Paragraph Essay

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