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Question Number 78134 by msup trace by abdo last updated on 14/Jan/20

$$findthevalueof{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{arctan\left(3x^2\right)}{x^2+4}dx$$

Commented by mathmax by abdo last updated on 14/Jan/20

$$letA={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{arctan\left(3x^2\right)}{x^2+4}dxandW\left(z\right)=\frac{arctan\left(3z^2\right)}{z^2+4}$$$$W\left(z\right)=\frac{arctan\left(3z^2\right)}{(z-2i)(z+2i)}\Rightarrow {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}W\left(z\right)dz=2i\pi Res(W,2i)$$$$=2i\pi \times \frac{\mid arctan\left(3{\left(2i\right)}^2\right)\mid }{4i}=\frac{\pi }{2}\mid arctan(-12)\mid =\frac{\pi }{2}arctan\left(12\right)\Rightarrow$$$$A=\frac{\pi }{2}arctan\left(12\right)$$

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