Question

The value of the integral
$ \int\limits^2_{1}e^{x} \left(log_{e} \,x + \frac{x+1}{x}\right)dx$ is

• A. $e^2(1 + log_e 2)$
• B. $e^2 - e$
• C. $e^2(1 + log_e 2) - e$
• D. $e^2 -e(1 + log_e 2)$

Answer 1

Answer: (C)

Let $/=\int_{1}^{2} e^{x}\left(\log _{\theta} x+\frac{x+1}{x}\right) d x$
$\Rightarrow I=\int_{1}^{2}\left(e^{x} \cdot \log _{e} x+e^{x}+\frac{e^{x}}{x}\right) d x$
$\Rightarrow I=\int_{1}^{2} e^{x} \log _{\theta} x d x+\int_{1}^{2} e^{x} d x+\int_{1}^{2} \frac{e^{x}}{x} d x$
$\Rightarrow I=\int_{1}^{2} e^{x} \log _{e} x d x+\left[e^{x}\right]_{1}^{2}+\left[e^{x} \log _{\theta} x_{1}^{2}\right.$
$-\int_{1}^{2} e^{x} \log _{\theta} x d x$
$\Rightarrow I=\left(e^{2}-e^{1}\right)+\left(e^{2} \log _{\theta} 2-0\right)$
$\Rightarrow e^{2}\left(1+\log _{\theta} 2\right)-e$