Question

Evaluate the definite integral. $$ \int_{0}^{2} \frac{x^{2}}{e^{x}} d x $$

Short answer

The evaluation of the definite integral given is \(8e^{-2}\).

Step-by-step solution

Identifying 'u' and 'dv'

To initiate integration by parts, it is needed to select 'u' and 'dv' judiciously from the integral. Here, let 'u' equals \(x^{2}\) and 'dv' equals \(e^{-x} dx\).

Calculating 'du' and 'v'

Now, calculate the derivatives and integrals involved by differentiating 'u' to get 'du' and integrating 'dv' to get 'v'. The derivative of 'u' i.e. 'du' equals \(2x dx\) and the integral of 'dv' i.e. 'v' equals \(-e^{-x}\).

Apply the integration by parts formula

Implementation of the integration by parts formula \(\int udv = uv - \int vdu\) now. This gives us \(-(x^{2}e^{-x}) - \int -2xe^{-x}dx \).

Second Application of Integration by Parts

The remaining integral \(\int -2xe^{-x}dx \) also requires integration by parts. Apply it again by letting 'u' equals \(x\) and 'dv' equals \(2e^{-x} dx\), then differentiate 'u' to get 'du' equals \(dx\) and integrate 'dv' to get 'v' equals \(-2e^{-x}\). After application, the expression becomes \(-x^{2}e^{-x} + 2(xe^{-x} - \int -2e^{-x} dx)\).

Final Integration and Applying Limits

Next, the remaining integral can be integrated in a straightforward manner. So, it leads to \(-x^{2}e^{-x} + 2(xe^{-x} + 2e^{-x})\). Now, apply the bounded limits from 0 to 2 to get \(-2^{2}e^{-2} + 4*2e^{-2} + 4e^{-2} - (-0^{2}e^{-0} + 0*0e^{-0} + 0)\).

Simplification

Finally, simplify to get the result. This results in: \(-4e^{-2} + 8e^{-2} + 4e^{-2} = 8e^{-2}\).

Key concepts

Integration by Parts

Integration by parts is a technique used to integrate products of functions that are not so straightforward to integrate directly. In calculus, it's a tool that allows us to break down complicated expressions into simpler ones. The formula, derived from the product rule of differentiation, states:
\[ \int u dv = uv - \int v du \]
In the exercise, selecting \( u = x^2 \) and \( dv = e^{-x} dx \) wisely is the first step; this is because their derivatives and integrals lead to simpler expressions. Following the formula, after finding \( du = 2x dx \) and \( v = -e^{-x} \), you substitute them back into the formula to progress toward the solution.

Why is the choice of 'u' and 'dv' important?

Picking 'u' and 'dv' properly is crucial because it can simplify the integration process significantly. Ideally, 'u' should be a function that becomes simpler upon differentiation, and 'dv' should be a function that does not become more complicated upon integration.

Common Pitfalls to Avoid

• Choosing 'dv' so that its integral becomes more complicated
• Forgetting to apply the antiderivative to 'v' after integrating 'dv'
• Skipping the distribution of negative signs when integrating-'uv' is as important as simplifying '\(\int v du\)'

Definite Integral Evaluation

Once the indefinite integration is performed using integration by parts or other techniques, the definite integral evaluation requires the application of limits to the resulting function. Specifically, after obtaining the antiderivative, you calculate its value at the upper limit of integration and subtract the value at the lower limit.

In the given exercise, after the integration process is complete, you apply the limits 0 and 2 to the resulting expression. This action gives you the definite integral's numerical value over the interval [0, 2].

Key Steps for Evaluation

To evaluate a definite integral:

• Find the indefinite integral or antiderivative
• Remember the '+ C' (the constant of integration) is not necessary in definite integrals
• Apply the upper limit to antiderivative and calculate its value
• Apply the lower limit to antiderivative and calculate its value
• Subtract the value at the lower limit from the value at the upper limit

But keep in mind, sometimes definite integrals may involve discontinuities or be improper; in such cases, the evaluation requires additional steps and consideration for convergence.

Calculus Step-by-Step Problem Solving

A step-by-step approach in calculus can be immensely beneficial for understanding and solving complex problems. This structured process allows you to dissect an issue into more manageable parts and solve them sequentially, leading to a clear path to the solution.

In the exercise above, a series of logical steps are followed to reach the answer. From identifying 'u' and 'dv' to applying the integration by parts formula multiple times, each step builds upon the previous. Simplifying the results at each stage ensures the problem doesn't get overwhelmingly complex.

Effective Steps in Problem Solving

• Identify which calculus concepts are applicable
• Break down the problem into smaller parts or steps
• Execute each step carefully, checking for errors
• Simplify at every opportunity to avoid complexity
• Keep a clear distinction between variables and constants

Adopting systematic problem solving reinforces your understanding of calculus concepts. It builds a necessary skill set that applies not just to academic exercises, but to real-world applications where complex problems are the norm.