Evaluate the following definite integrals. int_{1/8}^1frac{dx}{xsqrt{1+x^{2/3}}}

Evaluate the following definite integrals. int_{1/8}^1frac{dx}{xsqrt{1+x^{2/3}}}

Question
Integrals
asked 2020-12-14
Evaluate the following definite integrals.
\(\int_{1/8}^1\frac{dx}{x\sqrt{1+x^{2/3}}}\)

Answers (1)

2020-12-15
\(\text{Consider the integral:}\\
\(\int_{1/8}^1\frac{dx}{x\sqrt{1+x^{2/3}}}\\
\(\text{Apply u-substitution: }u=\sqrt{1+x^{2/3}}\\
\(\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\\
\(=3\cdot\int_{\frac{5^{1/2}}{2}}^{\sqrt{2}}\frac{1}{u^2-1}du\\
\(=3\cdot\int_{\frac{5^{1/2}}{2}}^{\sqrt{2}}\frac{1}{-(-u^2+1)}du\\
\(=3(-\int_{\frac{5^{1/2}}{2}}^{\sqrt{2}}\frac{1}{-u^2+1}du\\
\(=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{\sqrt5}{2}+1)+\ln(\frac{\sqrt5}{2}-1)}{2}\\
\(\text{Answer in terms of logarithms:}\\
\(\int_{1/8}^1\frac{dx}{x\sqrt{1+x^{2/3}}}=-3\cdot\frac{\ln(\sqrt{2}+1)-\ln(\sqrt{2}-1)-\ln(\frac{\sqrt5}{2}+1)+\ln(\frac{\sqrt5}{2}-1)}{2}\)
0

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