Consider the following equations: 

(1) ${\int }_0^{\mathrm{\infty }} \frac{1}{1+x^2}dx=\mathrm{\infty }$

(2) ${\int }_{-\pi /2}^{\pi /2} \cos x dx=2$

(3)  ${\int }_{-\pi /2}^{\pi /2} \sin \left|x\right|dx=2$

(4)  ${\int }_{-\pi /2}^{\pi /2} \sin x dx=-2$

Which of these are correct? 

 We take equation one-by-one 

(1) ${\int }_0^{\mathrm{\infty }} \frac{1}{1+x^2}dx={\left[{\tan }^{-1}x\right]}_0^{\mathrm{\infty }}$

$={\tan }^{-1}\left(\mathrm{\infty }\right)-{\tan }^{-1}\left(0\right)=\frac{\pi }{2}$

$\therefore$ ( 1) is incorrect 

(2) ${\int }_{-\pi /2}^{\pi /2} \cos xdx$

$\begin{array}{l}=2{\int }_0^{\pi /2} \cos xdx=2[\sin x{]}_0^{\pi /2}\\ =2\left[\sin \frac{\pi }{2}-{\left.0\right|}_{ }^{ }=2\times 1=2\right.\end{array}$

$\therefore$ (2) is correct