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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).

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Most popular questions from this chapter

If \(\int\left[\sqrt{\mathrm{x}} / \sqrt{\left. \left.\left(1-\mathrm{x}^{3 / 2}\right)\right] \mathrm{d} \mathrm{x}=\mathrm{P} \sqrt{\left(1-\mathrm{x}^{3 / 2}\right.}\right)+\mathrm{c} \text { then } \mathrm{P}=}\right.\) (a) \((4 / 3)\) (b) \((3 / 4)\) (c) \([(-4) / 3]\) (d) \([(-3) / 4]\)

If \(\mathrm{f}(\mathrm{x})=\cos \mathrm{x}-\cos ^{2} \mathrm{x}+\cos ^{3} \mathrm{x}-\cos ^{4} \mathrm{x}+\underline{\mathrm{x}}+\) then \(\int f(x) d x=\) \(+c\) (a) \(\tan (\mathrm{x} / 2)\) (b) \(x+\tan (x / 2)\) (b) \(x-(1 / 2) \tan (x / 2)\) (d) \(x-\tan (x / 2)\)

\(\int(x+4)(x+3)^{7} d x=\) (a) \(\left[(x+3)^{9} / 9\right]-\left[(x+3)^{8} / 8\right]\) (b) \(\left[\left\\{(x+3)^{8}(8 x+33)\right\\} / 72\right]\) (c) \(\left[\left\\{(x+3)^{8}(8 x+33)\right\\} / 72\right]\) (d) \(\left[(x+3)^{8} / 8\right]\)

\(\int\left[\mathrm{d} \mathrm{x} /\left\\{\mathrm{x}\left(\mathrm{x}^{\mathrm{n}}+1\right)\right\\}\right]=\mathrm{c}\) (a) \((1 / n) \log \left[\left(x^{n}+1\right) / x^{n}\right] \mid\) (b) \((1 / n) \log \left|\left[x^{n} /\left(x^{n}+1\right)\right]\right|\) (c) \((1 / n) \log \left|x^{n}+1\right|\) (d) \((1 / n) \log \left|\left[\left(x^{n}-1\right) / x^{n}\right]\right|\)

\(\int\left[\left\\{\left(x^{2}-1\right) d x\right\\} /\left\\{\left(x^{4}+3 x^{2}+1\right) \tan \left[\left(x^{2}+1\right) / x\right]\right\\}\right]=\ldots\) (a) \(\log \left|\tan ^{-1}\\{\mathrm{x}+(1 / \mathrm{x})\\}\right|\) (b) \(\log \left|\tan ^{-1}\\{\mathrm{x}-(1 / \mathrm{x})\\}\right|\) (c) \(\tan ^{-1}[\mathrm{x}+(1 / \mathrm{x})]\) (d) \(\tan ^{-1}[\mathrm{x}-(1 / \mathrm{x})]\)
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