Select all expressions matching the value of the definite integral int _(1)^e(x^2+1)/(x)dx (e^2)/(2) (e^2+1)/(52) e^2+1

To find the value of the definite integral $\int_{1}^{e}\frac{x^{2}+1}{x}dx$, we can simplify the integrand first.<br /><br />$\int_{1}^{e}\frac{x^{2}+1}{x}dx = \int_{1}^{e}(x+ \frac{1}{x})dx$<br /><br />Now, we can integrate term by term:<br /><br />$\int_{1}^{e}(x+ \frac{1}{x})dx = \int_{1}^{e}xd \int_{1}^{e}\frac{1}{x}dx$<br /><br />Integrating each term separately:<br /><br />$\int_{1}^{e}xdx = \left[\frac{x^2}{2}\right]_{1}^{e} = \frac{e^2}{2} - \frac{1}{2}$<br /><br />$\int_{1}^{e}\frac{1}{x}dx = \left[\ln|x|\right]_{1}^{e} = \ln(e) - \ln(1) = 1$<br /><br />Adding the results:<br /><br />$\int_{1}^{e}\frac{x^{2}+1}{x}dx = \frac{e^2}{2} - \frac{1}{2} + 1 = \frac{e^2}{2} + \frac{1}{2} = \frac{e^2 + 1}{2}$<br /><br />Therefore, the correct answer is $\frac{e^2 + 1}{2}$.