Question

If ${\int }_3^{10}\log \left[x\right]dx$ is equal to, where $\left[x\right]$ denotes greatest integer function

• A. $9+e$
• B. $15-e$
• C. $12$
• D. $9$

Answer 1

The correct option is D $9$

The value of

${\int }_3^{10}\log \left[x\right]dx$

$={\int }_3^4\log \left[x\right]dx+{\int }_4^5\log \left[x\right]dx+{\int }_5^6\log \left[x\right]dx+{\int }_6^7\log \left[x\right]dx+{\int }_7^8\log \left[x\right]dx+{\int }_8^9\log \left[x\right]dx+{\int }_9^{10}\log \left[x\right]dx$

$={\int }_3^4\log 3dx+{\int }_4^5\log 4dx+{\int }_5^6\log 5dx+{\int }_6^7\log 6dx+{\int }_7^8\log 7+{\int }_8^9\log 8dx+{\int }_9^{10}\log 9dx$

$=\log 3+\log 4+\log 5+\log 6+\log 7+\log 8+\log 9$

$=\log 181440$