question_answer

Evaluate \[\int_{0}^{1}{{{3}^{x-\,|x|}}dx.}\]

Answer:

Let \[l=\int_{0}^{1}{{{3}^{x-[x]}}dx}\] \[=\int_{0}^{1}{{{3}^{x-\,0}}dx}\]    \[[\because \,\,[x]=0,\,\,x\in (0,1)]\] \[=\left[ \frac{{{3}^{x}}}{\log \,\,3} \right]_{0}^{1}\]   \[\left[ \because \,\,\int{{{a}^{x}}dx=\frac{{{a}^{x}}}{\log \,\,a}} \right]\] \[=\left[ \frac{{{3}^{1}}}{\log \,\,3}-\frac{{{3}^{0}}}{\log 3} \right]\] \[=\frac{3}{\log \,\,3}-\frac{1}{\log 3}=\frac{2}{\log \,\,3}\]