Question

Evaluate: ${\int }_0^{\infty }\left[\frac{[x+1]}{e^{5x+2}}\right]dx$

where, $\left[x\right]$ denotes the greator integer less than or equal to '$x$'

• A. $0$
• B. $1$
• C. $\infty$
• D. $2$

Answer 1

The correct option is A $0$

${\int }_{\infty }^{\infty }\left[\frac{[x+1]}{e^{5x+2}}\right]dxletf\left(x\right)=\frac{[x+1]}{e^{x+2}}valueoff\left(0\right)=\frac{1}{e^2}\exists (0,1)f\left(0\right)=\frac{2}{e^3}\exists (0,1)f\left(0\right)=\frac{3}{e^4}\exists (0,1)andsoonforallvalueofxf\left(x\right)\exists (0,1)=\left[f\right(x\left)\right]=0\left[x\right]⟶G.1.F={\int }_0^{\infty }\left[f\right(x\left)\right]dx={\int }_0^{\infty }0dx=0$