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Chapter 11: Problem 896

\(\int \mathrm{e}^{4 \log \mathrm{x}}\left(\mathrm{x}^{5}+1\right)^{-1}=\underline{\mathrm{c}}\) (a) \((1 / 5) \log \left(\mathrm{x}^{4}+1\right)\) (b) \(-\log \left(\mathrm{x}^{4}+1\right)\) (c) \(\log \left(x^{4}+1\right)\) (d) \((1 / 5) \log \left(x^{5}+1\right)\)

Short Answer

Expert verified
(d) \(\frac{1}{5}\log\left(x^5 + 1\right)\)

Step by step solution

01

Simplify the integrand

Apply the logarithm properties of \(\exp\) and \(\log\). Since \(\mathrm{e}^{4\log\mathrm{x}} = \mathrm{x}^4\), we can rewrite the given integral as: \[\int \mathrm{x}^4\left(\mathrm{x}^{5}+1\right)^{-1}\,\mathrm{d}x\]
02

Integrate using substitution

Let us perform a substitution: \[u = x^5 + 1\] \[ \frac{\mathrm{d}u}{\mathrm{d}x} = 5x^4\] \[\mathrm{d}u = 5x^4\mathrm{d}x\] Substituting into the integral: \[\int \frac{1}{5}\cdot\frac{\mathrm{d}u}{u}\]
03

Integrating

Now we can easily integrate the expression: \[\frac{1}{5} \int \frac{\mathrm{d}u}{u}\] The integral of \(1/u \mathrm{d}u\) is \(\log(|u|) + C\), so our final expression will be: \[\frac{1}{5}\log(|u|) + C\]
04

Reverse substitution

Substitute the original variable \(x\) back in: \[\frac{1}{5}\log(x^5 + 1) + C\] Comparing the options given, it matches with option (d): \(\boxed{\text{(d) }\frac{1}{5}\log\left(x^5 + 1\right)}\)

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Most popular questions from this chapter

\(\int(x-1) e^{-x} d x=\square+c\)

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) \(\left[(-4025) /(2013)^{2}\right]\)

\(\int \sqrt{(1+\operatorname{cosec} x d x)}=\ldots\) (a) \(2 \sin ^{-1}(\sqrt{\cos } \mathrm{x})\) (b) \(2 \cos ^{-1}(\sqrt{\sin } \mathrm{x})\) (c) \(\sin ^{-1}(2 \sin x-1)\) (d) \(2 \cos ^{-1}(\sqrt{\cos x})\)

$$ \begin{aligned} &\int\left[(\mathrm{dx}) /\left\\{(\mathrm{x}-1)^{3 / 2}(\mathrm{x}-2)^{1 / 2}\right\\}\right]=\underline{\mathrm{c}}\\\ &\text { (a) } 2 \sqrt{[}(\mathrm{x}-1) /(\mathrm{x}-2)] \end{aligned} $$

If \(\int_{x \operatorname{cosec}^{2} x d x}=P \cdot x \cot x+Q \log |\sin x|+C\) then \(P+Q=\) (a) 1 (b) 2 (c) 0 (d) \(-1\)
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