91Ó°ÊÓ

Definite integrals calculate the area under a curve over a specific interval. They have clear boundaries, known as limits of integration, which indicate where to start and stop. For the exercise at hand, this interval is from 0 to 2, denoted as \(\int_{0}^{2}\).
When you evaluate a definite integral, you determine the net area between the curve and the x-axis, considering parts above the x-axis as positive and below as negative. The process results in a specific numerical value, which represents the accumulated value over the given interval. In this problem, by applying the definite integral over the function \(x e^{x}\), the solution should yield an exact number, providing insight into the behavior of the function across the interval from 0 to 2.

• Choose the function and limits of integration carefully.
• Apply limits after performing integration.
• Definite integrals always result in a finite number.

There are various integration techniques in calculus to find the integral of different functions. One such technique is **Integration by Parts**, useful when integrating products of functions. This involves breaking down a given integral into parts that are simpler to evaluate.
Integration by parts leverages the formula \(\int u \mathrm{d}v = uv - \int v \mathrm{d}u\), transforming an often difficult integral into a more manageable form. Choosing the appropriate functions \(u\) and \(\mathrm{d}v\) is crucial. An ideal approach is to select a \(u\) that simplifies upon differentiation and a \(\mathrm{d}v\) that is straightforward to integrate. In the given problem:

• **Chosen functions**: \(u = x\) and \(\mathrm{d}v = e^x \mathrm{d}x\).
• **Differentiation and Integration**: Differentiate \(u\) to get \(\mathrm{d}u\) and integrate \(\mathrm{d}v\) to find \(v\).
• **Apply the formula**: Substitute and simplify the integral.

Calculus problems often demand a blend of integration techniques and a strategic approach. When faced with problems that involve definite integrals, like \(\int_{0}^{2} x e^{x} \mathrm{d}x\), it is essential to:
- Break down the problem step by step.- Select the correct method, here it is the integration by parts.- Follow through with persistence and attention to detail.
In our scenario, the calculus problem required finding an integral for a product of polynomial and exponential functions over a set interval. It necessitated applying integration by parts to simplify and compute the integral efficiently. Once simplified, evaluate at the prescribed limits:

• Calculate the function's result at the upper limit.
• Subtract the function's result at the lower limit from the above.
• The difference is the solution, providing an exact form \(e^{2} - 1\).