Integral of x times e to the x using integration by parts
Evaluate the integral using integration by parts.
The task of evaluating the integral of x times e to the power of x is a classic problem illustrating the integration by parts technique. This technique is derived from the product rule for differentiation and offers a strategic way to handle integrals involving the product of two functions. The choice of which function to differentiate and which to integrate is often guided by the LIATE rule, which suggests starting with Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions in that order. For the given problem, x is chosen as the function to differentiate, and e raised to the x power is chosen to integrate, simplifying the process. Understanding integration by parts is crucial as it is widely applicable in solving integrals involving product functions, and comprehension of this method can significantly ease the tackling of complex integrals in advanced calculus.