Question

The value of ${\int }_1^e{10}^{{\log }_{e}x}dx$ is equal to

• A. $10{\log }_e(10e)$
• B. $\frac{10e-1}{{\log }_{e}10e}$
• C. $\frac{10e}{{\log }_{e}10e}$
• D. $(10e){\log }_e(10e)$
• E. $\frac{10e+1}{{\log }_{e}10e}$

Answer 1

Answer: (B)

Let $I={\int }_1^e{10}^{{\log }_{e}x}dx$ Again, let $I_1=\int {10}^{{\log }_{e}x}dx$
$\Rightarrow$ $I_1=x{.10}^{{\log }_{e}x}-\int x{.10}^{{\log }_{e}x}.\frac{{\log }_{e}10}{x}dx$
$\Rightarrow$ $I_1=x{10}^{{\log }_{e}x}-\int {10}^{{\log }_{e}x}{\log }_{e}10dx$
$\Rightarrow$ $(1+{\log }_{e}10)I_1=x{10}^{{\log }_{e}x}$
$\Rightarrow$ $I_1=\frac{x{.10}^{{\log }_{e}x}}{1+{\log }_{e}10}$
$\therefore$ $I={\left[\frac{x{.10}^{{\log }_{e}x}}{1+{\log }_{e}10}\right]}_1^e$
$=\left[\frac{10e-1}{1+{\log }_{e}10}\right]$
$=\frac{10e-1}{{\log }_{e}10e}$