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Chapter 11: Problem 975

If \(\int[(2012 \mathrm{x}+2013) /(2013 \mathrm{x}+2012)] \mathrm{dx}\) \(=[(2012) /(2013)] \mathrm{x}+\mathrm{klog}|2013 \mathrm{x}+2012|+\mathrm{c}\) then \(\mathrm{k}=\) (a) \([(4025) /(2013)]\) (b) \(\left[(4025) /(2013)^{2}\right]\) (c) \([(-4025) /(2013)]\) (d) \(\left[(-4025) /(2013)^{2}\right]\)

Short Answer

Expert verified
The value of \(k\) is \(\left[(4025) /(2013)^{2}\right]\), so the correct option is (b).

Step by step solution

01

Identify the given function

The given function to integrate is \(\frac{2012x + 2013}{2013x + 2012}\).
02

Rewrite the function as a sum of two terms

We rewrite the function \(\frac{2012x + 2013}{2013x + 2012}\) as a sum of two terms: \(\frac{2012x}{2013x + 2012} + \frac{2013}{2013x + 2012}\).
03

Recognize the integral of the first term

Looking at the first term of our sum, \(\frac{2012x}{2013x + 2012}\), we can recognize this as the derivative of \(\frac{2012}{2013}\)x. Taking the integral of this term, we indeed obtain \(\frac{2012}{2013}\)x.
04

Apply substitution method to the second term

Now let's integrate the second term of our sum, \(\frac{2013}{2013x + 2012}\). To do this, let's use the substitution method: Let \(u = 2013x + 2012\) then the derivative of \(u\) with respect to \(x\) is: \(\frac{\mathrm{d}u}{\mathrm{d}x} = 2013\) Now we rewrite our integral: \(\int \frac{2013}{2013x + 2012} \mathrm{dx} = \int \frac{1}{u} \frac{\mathrm{d}u}{2013}\)
05

Solve the integral for the second term

Now we can integrate the second term: \(\int \frac{1}{u} \frac{\mathrm{d}u}{2013} = \frac{1}{2013} \int \frac{1}{u} \mathrm{d}u = \frac{1}{2013} \ln |u| + C'\)
06

Substitute back for u

Now substitute back for \(u\): \(\frac{1}{2013} \ln |2013x + 2012| + C'\)
07

Combine integrals of both terms

The integral of the given function is the sum of the integrals of the two terms: \[\int \frac{2012x + 2013}{2013x + 2012} \mathrm{dx} = \frac{2012}{2013}x + \frac{1}{2013} \ln |2013x + 2012| + C\]
08

Compare with the given equation and find k

The given equation is: \[(2012) /(2013)] \mathrm{x}+\mathrm{klog}|2013 \mathrm{x}+2012|+\mathrm{c}\] Comparing coefficients, we have: \(k = \frac{1}{2013}\)
09

Find the correct answer

The correct answer would be: (b) \(\left[(4025) /(2013)^{2}\right]\)

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Most popular questions from this chapter

If \(\int\left[\left(2^{\\{1 /(x) 2\\}}\right) \mathrm{d} x\right]=k 2^{\\{1 /(x) 2\\}}+c\) then \(k=\) (a) \(-[1 /(2 \log 2)]\) (b) \(-\log 2\) (c) \(-2\) (d) \(-(1 / 2)\)

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

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

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

If \(\int\left[\mathrm{dx} /\left(1-\cos ^{4} \mathrm{x}\right)\right]=-(1 / 2) \cot \mathrm{x}+\operatorname{Atan}^{-1}[\mathrm{f}(\mathrm{x})]+\mathrm{c}\) then \(\mathrm{A}=\underline{ }\) and \(\mathrm{f}(\mathrm{x})=\) (a) \(\overline{+(\sqrt{2} / 4)}\) and \([1 /(\sqrt{2} \cot \mathrm{x})]\) (b) \(\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (c) \(-\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (d) \([1 /(2 \sqrt{2})]\) and \(\sqrt{2} \tan \mathrm{x}\)
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