Question

$\int_{0}^{\infty}\left\lbrack \frac{3}{x^{2} + 1} \right\rbrack dx$ is equal to [.] denotes the greatest integer

function)

• A. $\sqrt{2}$
• B. $\sqrt{2}$+ 1
• C. 3/$\sqrt{2}$
• D. Infinite

Answer 1

Answer: (C)

3/$\sqrt{2}$

Answer 2

Here 0 <$\frac{3}{x^{2} + 1} \leq 3$, $\frac{3}{x^{2} + 1} = 2$

⇒ x = $\frac{1}{\sqrt{2}}$ and $\frac{3}{x^{2} + 1}$ = 1 ⇒ x = $\sqrt{2}$

⇒ I =$\int_{0}^{1/\sqrt{2}}{2dx}$+$\int_{1/\sqrt{2}}^{\sqrt{2}}{1.dx}$ + $\int_{\sqrt{2}}^{\infty}{0 ⥄ dx}$ = $\sqrt{2}$+ $\sqrt{2} - \frac{1}{\sqrt{2}}$

= 2$\sqrt{2} - \frac{1}{\sqrt{2}}$ =$\frac{3}{\sqrt{2}}$