Questions: The definite integral ∫ from 1 to e of (x+1) log x dx equals Select one: e+1-2 log 1+c, c ∈ R e+1 1/4(e^2+5) 1/4(e^2+5)+c, c ∈ R e^2+5

Solution Steps

We need to evaluate the definite integral

\[ \int_{1}^{e}(x+1) \log x \, dx. \]

Using integration by parts, we let

\[ u = \log x \quad \text{and} \quad dv = (x + 1) \, dx. \]

Then, we find

\[ du = \frac{1}{x} \, dx \quad \text{and} \quad v = \frac{x^2}{2} + x. \]

Applying the integration by parts formula

\[ \int u \, dv = uv - \int v \, du, \]

we compute the integral and evaluate it from 1 to \(e\). The result of the definite integral is

\[ \frac{1}{4}(e^2 + 5). \]

Final Answer