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Chapter 11: Problem 918

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

Short Answer

Expert verified
\(c\)

Step by step solution

01

Break down the given integral

The given integral equation is \( \int [\log (\log x) + \frac{1}{\log x}] \, dx \). Break this down into two separate integrals: \( \int \log (\log x)\, dx + \int \frac{1}{\log x} \, dx \).
02

Integration of the first term

The first integral involves the logarithm of a logarithmic function. The integration by parts method (which states that \( \int u \, dv = u v - \int v \, du\)) can handle this kind of problem. Start by letting \( u = \log (\log x)\) and \( dv = dx\). Then calculate \( du = \frac{1}{x \, \log x} \, dx \) and \( v = x \). So according to the integration by parts rule, \( \int u \, dv = u v - \int v \, du = \log(\log x) \cdot x - \int x \cdot \frac{1}{x \, \log x} \, dx = x \, \log(\log x) - \int \frac{1}{\log x} \, dx \).
03

Integration of the second term

The second integral is \( \int \frac{1}{\log x} \, dx \). We can use the direct integration method for this. It's known that the integral of \( \frac{1}{\log x} \, dx \) is \( x \, \log(\log x) \).
04

Combine the results of Step 2 and Step 3

By integrating the terms separated in Step 1, we get \( x \, \log(\log x) - x \, \log(\log x) \) which simplifies to 0. However, as this is an indefinite integral, we have to add the integration constant. So the final antiderivative of the function is \( c \).
05

Match the result with the provided choices

The final antiderivative obtained doesn't match with any of the answer choices given. So, the correct answer seems to be missing from the provided options. The steps followed here have shown how to integrate the function given in the problem using integration methods for logarithmic functions.

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Most popular questions from this chapter

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

If \(\int\left[\left\\{x^{2012} \tan ^{-1}\left(x^{2012}\right)\right\\} /\left(1+x^{4024}\right)\right] d x=k \tan ^{-1}\left(x^{2012}\right)+c\) (a) \([1 /(2012)]\) (a) \(-[1 /(2012)]\) (c) \([1 /(4024)]\) (d) \(-[1 /(4024)]\)

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

\(I=\int\left[\left(x^{3} d x\right) /\left\\{\sqrt{\left. \left.\left(1+x^{8}\right)\right\\}\right]}\right.\right.\) (a) \(\log \mid x^{4}+\sqrt{\left(1+x^{8}\right) \mid+c}\) (b) \(\log \mid \sqrt{\left(x^{8}+1\right) \mid+c}\) (c) \((1 / 4) \log \mid \mathrm{x}^{4}+\sqrt{\left(1+\mathrm{x}^{8}\right) \mid+\mathrm{c}}\) (d) none of these

\(\int \mathrm{e}^{2 \mathrm{x}+\log \mathrm{x}} \mathrm{dx}=\) (a) \((1 / 4)(2 \mathrm{x}-1) \mathrm{e}^{2 \mathrm{x}}\) (b) \((1 / 2)(2 \mathrm{x}-1) \mathrm{e}^{2 \mathrm{x}}\) (c) \((1 / 4)(2 \mathrm{x}+1) \mathrm{e}^{2 \mathrm{x}}\) (d) \((1 / 4)(2 \mathrm{x}+1) \mathrm{e}^{2 \mathrm{x}}\)
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