Question

Evaluate the following definite integrals. $$ \int_{-1}^{1} 4 x e^{x^{2}} d x $$

Short answer

Question: Evaluate the definite integral $\int_{-1}^{1} 4 x e^{x^2} dx$. Answer: $2e - 2 \frac{1}{e}$.

Step-by-step solution

Identify the integrand and bounds of integration

The integral we are given is $$ \int_{-1}^{1} 4 x e^{x^{2}} d x $$ The integrand is \(4xe^{x^2}\), and the bounds of integration are from -1 to 1.

Find the antiderivative of the integrand

To find the antiderivative of the integrand, we will use a substitution method. Let \(u = x^2\). Then, we have \(\frac{du}{dx} = 2x\), which means \(dx = \frac{du}{2x}\). Now, we can rewrite the integral in terms of \(u\): $$ \int_{-1}^{1} 4x e^{x^2} dx = \int 4 e^u \cdot \frac{du}{2x} $$ The integral becomes: $$ 2\int e^u du $$ The antiderivative of \(e^u\) is simply \(e^u\). Therefore, the antiderivative of the integrand is: $$ 2e^u + C $$ Now, we need to convert back to the original variable \(x\). Since \(u = x^2\), we have: $$ 2e^{x^2} + C $$ This is the antiderivative we will use to evaluate the definite integral.

Evaluate the antiderivative at the bounds of integration

Now that we have found the antiderivative, we can evaluate it at the bounds of integration: $$ (2e^{1^2} + C) - (2e^{-1^2} + C) $$ Since the +C terms cancel out, we are left with: $$ 2e^1 - 2e^{-1} $$

Simplify the result

Finally, we simplify the result: $$ 2(e - \frac{1}{e}) = 2e - 2 \frac{1}{e} $$ Thus, the value of the given definite integral is: $$ \int_{-1}^{1} 4 x e^{x^{2}} d x = 2e - 2 \frac{1}{e} $$

Key concepts

Integration by Substitution

Integration by substitution is a powerful technique used to simplify complex integrals. It's similar to solving a difficult puzzle by changing the pieces' arrangements to make it easier. Here's how it works.

When we have an integral in the form of a complicated function, we first identify a part of this function that can be replaced with a simpler variable. This substitution helps us transform the integral into an easier form.

• In the original exercise, we let \(u = x^2\).
• This new substitution implies \(\frac{du}{dx} = 2x\), so \(dx = \frac{du}{2x}\).

By substituting \(u\) into the integral, the equation simplifies significantly. This simplified form can often be integrated more easily, allowing us to proceed further in solving the problem.
Remember, after integrating with respect to \(u\), always convert back to the original variable \(x\), so that the final result reflects our initial problem.

Antiderivative

The antiderivative, also known as the indefinite integral, is the function that we differentiate to get our original integral function. It is essentially the reverse process of differentiation.

• For the given function in the exercise, the task is to find the antiderivative of \(4x e^{x^2}\).
• Using substitution, the function became \(2 \int e^u \, du\).
• The antiderivative of \(e^u\) is simply \(e^u\).

Once we recognize this, integrating becomes straightforward. The antiderivative of \(2e^u\) thus is \(2e^u + C\), where \(C\) is the constant of integration.

After finding the antiderivative in terms of \(u\), we use our previous substitution \(u = x^2\) to revert back to the original variable, giving us \(2e^{x^2} + C\). This result is crucial for further evaluating the definite integral.

Evaluating Integrals

Once the antiderivative of a function is found, the next step is to evaluate the integral, especially when dealing with definite integrals. This means we use the Fundamental Theorem of Calculus which connects differentiation and integration.

With definite integrals, we deal with specific bounds, or limits, of integration. The process involves:

• Substituting the upper bound into the antiderivative.
• Substituting the lower bound into the antiderivative.
• Subtracting the value obtained from the lower bound from the upper bound's result.

In our exercise, evaluating from -1 to 1 involves computing \((2e^{1^2}) - (2e^{-1^2})\). The constant \(C\) cancels out since \(C - C = 0\). This step confirms the evaluated integral, resulting in \(2e - 2 \frac{1}{e}\), which represents the area under the curve of the original function from -1 to 1.

Bounds of Integration

In definite integrals, bounds of integration are the limits between which we calculate the integral. They are like the start and end points on a journey, dictating where the integration process begins and finishes.

The bounds crucially define the portion of the area under the curve we are interested in measuring.

• For our given problem, the bounds are \(-1\) to \(1\).
• These are the points at which we evaluate the antiderivative, plugging them into the final expression.

It's important to pay close attention to these bounds because they affect the results significantly. They tell us whether the region under the curve spans multiple quadrants or is confined to a specific section.

Moreover, when evaluating, you must ensure that the function is correctly simplified at both bounds to achieve the correct solution.