Question

Solve ${\int }_0^{\left[1\right]}\frac{3^x}{3^{\left[x\right]}}dx$, where [.] denotes greatest integer function.

Answer 1

${\int }_0^{\left[1\right]}\frac{3^x}{3^{\left[x\right]}}dx$

${\int }_0^{\log 3^2}{3}^{x}dx+{\int }_{\log 3^2}^1\frac{3^{x}dx}{2}$

$={\left[\frac{3^x}{l{n}^3}\right]}_0^{\log 3^2}+{\left[\frac{3^x}{2ln3}\right]}_{\log 3^2}^1$

$=\frac{2}{ln3}-\frac{1}{ln3}+\frac{3}{2ln3}-\frac{1}{ln3}$

$=\frac{7}{2ln3}-\frac{2}{ln3}=\frac{3}{2ln3}$ Ans