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Question Number 148884 by 0731619 last updated on 01/Aug/21

Answered by aleks041103 last updated on 03/Aug/21

$${\int }_0^{\mathrm{\infty }}\frac{dx}{a+x^{13}}=\frac{1}{a}{\int }_0^{\mathrm{\infty }}\frac{dx}{1+{\left(\frac{x}{a^{1/13}}\right)}^{13}}=$$$$=\frac{a^{1/13}}{a}{\int }_0^{\mathrm{\infty }}\frac{d\left(x/a^{1/13}\right)}{1+{\left(x/a^{1/13}\right)}^{13}}$$$$ifa>0,thenwesubstitute$$$$u=x/a^{1/13}$$$$Then:$$$${\int }_0^{\mathrm{\infty }}\frac{dx}{a+x^{13}}=a^{-12/13}{\int }_0^{\mathrm{\infty }}\frac{dx}{1+x^{13}}$$$$\Rightarrow \frac{{\int }_0^{\mathrm{\infty }}\frac{dx}{1+x^{13}}}{{\int }_0^{\mathrm{\infty }}\frac{dx}{13+x^{13}}}={13}^{12/13}$$

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