**Problem Statement:**Evaluate exactly:\[ \int_{0}^{\ln 4} xe^x \, dx \]---**Additional Notes:**- The problem is displayed in a word processor-like interface with formatting options.- Specifically, the problem asks to evaluate an integral with limits from 0 to \(\ln 4\). - The integrand is \(xe^x\).---This kind of problem is typically approached using integration by parts. To evaluate this integral:Let \( u = x \) and \( dv = e^x \, dx \).Then, the corresponding \( du = dx \) and \( v = e^x \).Using the integration by parts formula, \(\int u \, dv = uv - \int v \, du\), we proceed as follows:\[ \int_{0}^{\ln 4} xe^x \, dx = \left. x e^x \right|_{0}^{\ln 4} - \int_{0}^{\ln 4} e^x \, dx \]\[ \left. x e^x \right|_{0}^{\ln 4} = (\ln 4) e^{\ln 4} - 0 \cdot e^0 \]\[ = (\ln 4) \cdot 4 - 0 \]\[ = 4 \ln 4 \]Now, evaluating \(\int_{0}^{\ln 4} e^x \, dx \):\[ \int e^x \, dx = e^x \]\[ \left. e^x \right|_{0}^{\ln 4} = e^{\ln 4} - e^0 \]\[ = 4 - 1 = 3 \]Therefore,\[ \int_{0}^{\ln 4} xe^x \, dx = 4 \ln 4 - 3 \]Given the above steps, the exact value of the integral is:\[ \boxed{4 \ln 4 - 3} \]---The text also includes formatting tools, suggesting it was typed in an editable document where formatting (such as font type and size) can be adjusted. This section is not relevant to solving the integral but indicates how the problem might have been presented or inputted into the system.

Name: **Problem Statement:**Evaluate exactly:\[ \int_{0}^{\ln 4} xe^x \, dx
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: **Problem Statement:**Evaluate exactly:\[ \int_{0}^{\ln 4} xe^x \, dx \]---**Additional Notes:**- The problem is displayed in a word processor-like interface with formatting options.- Specifically, the problem asks to evaluate an integral with limit