 # Evaluate the following definite integrals. int_{1/8}^1frac{dx}{xsqrt{1+x^{2/3}}} Jaya Legge 2020-12-14 Answered
Evaluate the following definite integrals.
${\int }_{1/8}^{1}\frac{dx}{x\sqrt{1+{x}^{2/3}}}$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it wornoutwomanC

$\text{Consider the integral:}\phantom{\rule{0ex}{0ex}}$${\int }_{1/8}^{1}\frac{dx}{x\sqrt{1+{x}^{2/3}}}\phantom{\rule{0ex}{0ex}}$

${\int }_{1/8}^{1}\frac{dx}{x\sqrt{1+{x}^{2/3}}}={\int }_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\frac{3}{{u}^{2}-1}du\phantom{\rule{0ex}{0ex}}$
$=3\cdot {\int }_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\frac{1}{{u}^{2}-1}du\phantom{\rule{0ex}{0ex}}$
$=3\cdot {\int }_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\frac{1}{-\left(-{u}^{2}+1\right)}du\phantom{\rule{0ex}{0ex}}$
$=3-{\int }_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\frac{1}{-{u}^{2}+1}du\phantom{\rule{0ex}{0ex}}$
$=3\left(-\left[\frac{\mathrm{ln}|u+1|}{2}-\frac{\mathrm{ln}|u-1|}{2}{\right]}_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\right)\phantom{\rule{0ex}{0ex}}$
$=-3\left[\frac{1}{2}\left(\mathrm{ln}|u+1|-\mathrm{ln}|u-1|\right){\right]}_{\frac{{5}^{1/2}}{2}}^{\sqrt{2}}\phantom{\rule{0ex}{0ex}}$
$=-3\cdot \frac{\mathrm{ln}\left(\sqrt{2}+1\right)-\mathrm{ln}\left(\sqrt{2}-1\right)-\mathrm{ln}\left(\frac{\sqrt{5}}{2}+1\right)+\mathrm{ln}\left(\frac{\sqrt{5}}{2}-1\right)}{2}\phantom{\rule{0ex}{0ex}}\text{Answer in terms of logarithms:}\phantom{\rule{0ex}{0ex}}$
${\int }_{1/8}^{1}\frac{dx}{x\sqrt{1+{x}^{2/3}}}=-3\cdot \frac{\mathrm{ln}\left(\sqrt{2}+1\right)-\mathrm{ln}\left(\sqrt{2}-1\right)-\mathrm{ln}\left(\frac{\sqrt{5}}{2}+1\right)+\mathrm{ln}\left(\frac{\sqrt{5}}{2}-1\right)}{2}$