"Evaluate the following definite integrals. ∫1/81dxx1+x2/3" was answered by our experts

Jaya Legge

Answered question

2020-12-14

Evaluate the following definite integrals.

\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1 + x^{2 / 3}}}

Answer & Explanation

wornoutwomanC

Skilled2020-12-15Added 81 answers

$Consider the integral:$ $\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1 + x^{2 / 3}}}$
$Apply u-substitution: u = \sqrt{1 + x^{2 / 3}}$
$\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1 + x^{2 / 3}}} = \int_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}} \frac{3}{u^{2} - 1} d u$
$= 3 \cdot \int_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}} \frac{1}{u^{2} - 1} d u$
$= 3 \cdot \int_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}} \frac{1}{- (- u^{2} + 1)} d u$
$= 3 - \int_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}} \frac{1}{- u^{2} + 1} d u$
$= 3 (- [\frac{\ln | u + 1 |}{2} - \frac{\ln | u - 1 |}{2}]_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}})$
$= - 3 [\frac{1}{2} (\ln | u + 1 | - \ln | u - 1 |)]_{\frac{5^{1 / 2}}{2}}^{\sqrt{2}}$
$= - 3 \cdot \frac{\ln (\sqrt{2} + 1) - \ln (\sqrt{2} - 1) - \ln (\frac{\sqrt{5}}{2} + 1) + \ln (\frac{\sqrt{5}}{2} - 1)}{2} Answer in terms of logarithms:$
$\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1 + x^{2 / 3}}} = - 3 \cdot \frac{\ln (\sqrt{2} + 1) - \ln (\sqrt{2} - 1) - \ln (\frac{\sqrt{5}}{2} + 1) + \ln (\frac{\sqrt{5}}{2} - 1)}{2}$

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