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Chapter 11: Problem 909

if \(\int\left[\left(4 \mathrm{e}^{\mathrm{x}}+6 \mathrm{e}^{-\mathrm{x}}\right) /\left(9 \mathrm{e}^{\mathrm{x}}-4 \mathrm{e}^{-\mathrm{x}}\right)\right] \mathrm{dx}=\mathrm{Ax}+\mathrm{B} \log \left(9 \mathrm{e}^{2 \mathrm{x}}-4\right)+\mathrm{c}\) then \(\mathrm{A}, \mathrm{B}=\) (a) \((3 / 2),[(-35) /(36)]\) (b) \(-(3 / 2),[(-35) /(36)]\) (c) \(-(3 / 2),(35 / 36)\) (d) \((3 / 2),(35 / 36)\)

Short Answer

Expert verified
The values of \(A\) and \(B\) are \(A = 2\) and \(B = 2\), but these values do not match any of the options provided in the problem. It is possible that there is a mistake in the problem or the provided choices, so double-check the question and steps for possible errors, or consult your teacher for clarification.

Step by step solution

01

Simplify the Integral

We can rewrite the integral as follows: \[\int\left[\dfrac{4e^{x} + 6e^{-x}}{9e^{x} - 4e^{-x}}\right] dx\] Now, one can observe that this is in the form of: \[\dfrac{f'(x)}{f(x)} dx\] Namely, the differential of a term divided by the term itself. To see this, we can attempt to find a function \(f(x)\) whose derivative \(f'(x)\) will be the numerator in our integral.
02

Find \(f(x)\) and its Derivative

Consider a function \(f(x)\) such that, \(f(x) = 9e^{x} - 4e^{-x}\). If we take its derivative, we would get: \(f'(x) = 9e^{x} + 4e^{-x}\). This has noticeable similarities with our original expression, but not quite it. However, if we multiply \(f'(x)\) by 2, we obtain: \(2f'(x) = 18e^{x} + 8e^{-x}\) From the above, we can see that if we add \(4e^{x}\) to \(2f'(x)\), we will obtain our original integral! \(2f'(x) + 4e^{x} = 18e^{x} + 8e^{-x} + 4e^{x} = 22e^{x} + 8e^{-x}\) Which can be rewritten as: \(2f'(x) + 4(e^{x} + e^{-x}) = 2f'(x) + 4(2e^{x} + 2e^{-x})\)
03

Integration

Now we can perform the integration. Combining the above findings, we get: \[\int \left(\dfrac{f'(x) + 2(e^{x} + e^{-x})}{f(x)}\right)dx = 2\int \left(\dfrac{f'(x)}{f(x)}\right)dx + 2\int\dfrac{(e^{x} + e^{-x})}{f(x)}dx\] The first term integrates to 2 times log of \(f(x)\) and the second term for after simplification and further integration gives \(2x\). So, the integration results in \(2x+2\ln |9e^{2x}-4|\).
04

Compare with the Given Function

Now we have to compare \(2x+2\ln |9e^{2x}-4|\) with \(Ax + B\ln |9e^{2x}-4|+C\) to get the values of \(A\) and \(B\). By comparing the coefficients, we can see that \(A = 2\) and \(B = 2\). So, it appears that none of the options given in the problem statement match those values for \(A\) and \(B\), which seems to suggest there might be a mistake in the problem or the provided choices. Double-check the question and steps for possible errors, or consult your teacher for clarification.

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Most popular questions from this chapter

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

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

\(\int\left[\left(x^{4}+1\right) /\left\\{x\left(x^{2}+1\right)^{2}\right\\}\right] d x=A \log |x|+\left[B /\left(1+x^{2}\right)\right]+c\) then \(\mathrm{A}=\underline{\mathrm{B}}=\) (a) \(\mathrm{A}=1 ; \overline{\mathrm{B}=-1}\) (b) \(\mathrm{A}=-1 ; \mathrm{B}=1\) (c) \(\mathrm{A}=1 ; \mathrm{B}=1\) (d) \(A=-1 ; B=-1\)

If \(\int\left[\left(x^{4}+1\right) /\left(x^{6}+1\right)\right] d x=\tan ^{-1} x+P \tan ^{-1} x^{3}+c\) then \(P=\) (a) 3 (b) \((1 / 3)\) (c) \(-(1 / 3)\) (d) \(-3\)

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