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Question Number 166280 by Eulerian last updated on 17/Feb/22

$$\mathrm{Solve}\mathrm{for}\mathrm{the}\mathrm{exact}\mathrm{value}\mathrm{of}{\int }_0^{\mathrm{\infty }}\sin ({\mathrm{x}}^2+{\mathrm{x}}^{-2})\mathrm{dx}$$

Answered by phanphuoc last updated on 17/Feb/22

$$putx=1/t\to dx=-dt/t^2$$$$i={\int }_{\mathrm{\infty }}^0-sin(t^{-2}+t^2)dt/t^2$$$$2i={\int }_0^{\mathrm{\infty }}(1+1/x^2)sin(x^2+x^{-2})dx=$$$${\int }_0^{\mathrm{\infty }}(1+1/x^2)sin({(x-1/x)}^2-2)dx$$$$u=x-1/x\to du=(1+1/x^2)dx$$$$i=1/2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}sin(u^2-2)du=1/2{\left(\pi /2\right)}^{1/2}(cos2-sin2)$$$$$$

Commented by Eulerian last updated on 18/Feb/22

$$\mathrm{No}\mathrm{sir},\mathrm{try}\mathrm{to}\mathrm{recheck}\mathrm{your}\mathrm{work}!$$

Commented by MJS_new last updated on 18/Feb/22

$$\mathrm{I}\mathrm{get}\mathrm{the}\mathrm{same}\mathrm{with}(\cos 2+\sin 2)$$

Commented by Eulerian last updated on 19/Feb/22

$$\mathrm{Yeah}\mathrm{the}\mathrm{correct}\mathrm{answer}\mathrm{is}\left(\frac{\cos 2+\sin 2}{2}\right)\sqrt{\frac{\pi }{2}}$$

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