Question

${\int }_0^{\left\{x\right\}}\frac{3^x}{3^{\left\{x\right\}}}dx$, where $[\cdot ]$ denotes greatest integer function.

Answer 1

Given : ${\int }_0^{\left\{x\right\}}\frac{3^x}{3^{\left\{x\right\}}}dx$

$I={\int }_0^{\left\{x\right\}}{3}^{x-\left\{x\right\}}dx={\int }_0^{\left\{x\right\}}{3}^{\left[x\right]}dx$

We know $x$ lies between $0and-1$

And $\left[x\right]$ for $0to1$ is $0$

$I={\int }_0^{\left\{x\right\}}{3}^{\left[x\right]}dx={\int }_0^1{3}^{0}dx={\int }_0^{1}1dx=1$

Hence the correct answer is $1$