Question

The value of the integral $I=\displaystyle \int _{1}^{2}t^{\left[\left\{t\right\}\right] + t}\left(1 + ln t\right) \, dt$ is equal to

( $\left[.\right]$ and $\left\{.\right\}$ denotes the greatest integer and fractional part function respectively)

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (D)

As $\left[\left\{t\right\}\right]=0$ , we get,
$I=\displaystyle \int _{1}^{2}t^{t}\left(1 + ln t\right)dt$
$=\displaystyle \int _{1}^{2} d \left(t^{t}\right)$
$=\left[t^{t}\right]_{1}^{2}$
$=2^{2}-1^{1}=3$