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Question Number 196965 by sonukgindia last updated on 05/Sep/23

Answered by qaz last updated on 05/Sep/23

$${\int }_0^{\mathrm{\infty }}\frac{dx}{\sqrt{x}(1+x)(1+\sqrt[4]{x})}\stackrel{x\to x^4}{=}{\int }_0^{\mathrm{\infty }}\frac{4xdx}{(1+x^4)(1+x)}$$$$\stackrel{x\to \frac{1}{x}}{=}{\int }_0^{\mathrm{\infty }}\frac{4x^{2}dx}{(1+x^4)(1+x)}={\int }_0^{\mathrm{\infty }}\frac{2xdx}{1+x^4}=\frac{\pi }{2}$$

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