**B.**\[ \int_{1}^{e} x^2 \ln x \, dx \]This integral represents the definite integral of the function \(x^2 \ln x\) from the lower limit of 1 to the upper limit \(e\), where \(e\) is the base of the natural logarithm, approximately 2.718. This expression involves integration of a product of functions, which may require techniques such as integration by parts for its evaluation. The integral can be used to find the area under the curve of the function \(x^2 \ln x\) between the specified limits on the x-axis. 1. Use Integration by parts to solve the following integrals. Show all your work, including the steps to find all functions and their derivatives or antiderivatives in the integration by parts steps, and all integrals. Remember, sometimes you must use algebra to solve when seeing a pattern in integration by parts.

Given:- Use Integration by parts to solve the following integrals. Show all your work, including the steps to find all functions and their derivatives or anti derivatives in the integration by parts steps, and all integrals. Remember, sometimes you must use algebra to solve when seeing a pattern in integration by parts. ∫1ex2ln xdx Solution:- ∫1ex2ln xdx= x33ln x 1e - ∫1ex33dx =e33-0 - x391e⇒∫1ex2ln x dx = 2e39+19