Calculating the Integral of x^2 e^(-x^2) dx

Calculating the Integral $\int {�}^2{�}^{-{�}^2}��$

Integration is a fundamental concept in mathematics that plays a crucial role in finding the area under curves, solving differential equations, and analyzing various phenomena across multiple fields. While many integrals can be calculated using basic rules and functions, others present more significant challenges. The integral $\int {�}^2{�}^{-{�}^2}�$∫x is one such example. In this comprehensive article, we will delve into the step-by-step process of evaluating this integral.

Understanding the Integral

At first glance, the integral $\int {�}^2{�}^{-{�}^2}��$ appears intricate. It combines a polynomial function, ${�}^2$, with an exponential function, ${�}^{-{�}^2}$. Unlike simpler integrals that can be evaluated using elementary functions like polynomials, exponentials, or trigonometric functions, this integral lacks a closed-form antiderivative expressible in terms of such functions. Consequently, to solve it, we must employ a specialized technique known as substitution.

Step 1: Substitution

The initial step in tackling this integral is choosing an appropriate substitution. In this instance, we opt for $�=-{�}^2$. The selection of $�$ should aim to simplify the integral or make it more manageable. To proceed, we need to determine $��$ in terms of $��$. By differentiating $�$ with respect to $�$, we obtain $��=-2���$, implying that $��=-\frac{1}{2}��$.

Step 2: Changing Variables

With the substitution in place, we proceed to rewrite the original integral in terms of $�$:

$$\int {�}^2{�}^{-{�}^2}��=\frac{1}{2}\int 2�\cdot �{�}^{-{�}^2}��=-\frac{1}{2}\int {�}^{�}��$$

Here, we have expressed $��$ in terms of $��$ and simplified the integral, transforming it into a more manageable form.

Step 3: Computing the New Integral

Now, we are confronted with the integral $-\frac{1}{2}\int {�}^{�}��$. This integral is considerably simpler to evaluate compared to the original one. The integration of ${�}^{�}$ with respect to $�$ is a straightforward application of integration rules:

$$-\frac{1}{2}\int {�}^{�}��=-\frac{1}{2}{�}^{�}+{�}_{�}$$

Here, ${�}_{�}$ represents the constant of integration with respect to the variable $�$.

Step 4: Reverting to the Original Variable

Our ultimate goal is to express the result in terms of the original variable, $�$, rather than $�$. To achieve this, we must revert the substitution $�=-{�}^2$ back to $�$. Recall that $�=-{�}^2$, so ${�}^{�}={�}^{-{�}^2}$. Substituting this back into our result:

$$-\frac{1}{2}{�}^{�}+{�}_{�}=-\frac{1}{2}{�}^{-{�}^2}+{�}_{�}$$

Now, we have successfully expressed the result of the integral in terms of $�$.

Final Result

The final result of the integral $\int {�}^2{�}^{-{�}^2}��$ is as follows:

$$-\frac{1}{2}{�}^{-{�}^2}+{�}_{�}$$

In this expression, ${�}_{�}$ signifies the constant of integration introduced during the calculation with respect to $�$. Depending on the specific context or problem you are working on, this constant can vary.

Conclusion

In conclusion, the integral $\int {�}^2{�}^{-{�}^2}��$ presents a formidable challenge due to its inability to be solved using elementary functions. However, by employing the technique of substitution, we successfully transformed it into a more manageable form and expressed the result in terms of a simpler integral involving the exponential function. The final answer, $-\frac{1}{2}{�}^{-{�}^2}+{�}_{�}$, provides a solution to this integral, enabling us to calculate its value for various applications in mathematics, physics, and engineering.

This example underscores the power of mathematical techniques in addressing complex problems and finding solutions, even when closed-form expressions are not readily available. The process of substitution, as demonstrated here, is a valuable tool in the mathematician's toolkit, allowing us to tackle a wide range of integrals and mathematical challenges.