2. Evaluate the definite integral from $1$ to $e$: ${\int }_1^{e}xln\left(x\right)dx$.

• A. $\frac{e^2-1}{2}$
• B. $\frac{e^2-1}{4}$
• C. $e^2-1$
• D. $e-1$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the integral involves the product of x and ln(x). This suggests using integration by parts, which is given by the formula: ∫u dv = uv - ∫v du.
2. Step 2: Choose u = ln(x) and dv = x dx. Then, compute du = (1/x) dx and v = (x^2)/2 (since the integral of x dx is (x^2)/2).
3. Step 3: Substitute into the integration by parts formula: ∫x ln(x) dx = uv - ∫v du = ln(x) * (x^2)/2 - ∫((x^2)/2) * (1/x) dx.
4. Step 4: Simplify the remaining integral: ∫((x^2)/2) * (1/x) dx = ∫x/2 dx = (1/2) * ∫x dx. The integral of x dx is (x^2)/2, so this becomes (1/2) * (x^2)/2 = (x^2)/4.
5. Step 5: Combine the results and evaluate the definite integral from 1 to e. Substitute the limits into the expression obtained after integration by parts: [(ln(x) * (x^2)/2 - (x^2)/4)] evaluated from x = 1 to x = e.