What substitution should be used to compute the integral ∫ x^4 e^(-x^2) dx?

To evaluate the integral ∫ x^4 e^(-x^2) dx effectively, the ideal substitution involves relating the exponent of the exponential function to the variable x. The expression e^(-x^2) suggests that the substitution should simplify the power of x and transform the integral into a more manageable form.

Choosing the substitution u = -x^2 is advantageous because it directly impacts the term in the exponent of e^(-x^2). When differentiating this substitution, you find that du = -2x dx, which allows us to express dx in terms of du and x. Additionally, this substitution changes the e^(-x^2) into e^u, further simplifying the integral.

To express x^4 in terms of u, we may notice that x^2 = -u, leading to x^4 = (x^2)^2 = (-u)^2 = u^2. Thus, all components of the integral can be expressed in terms of u.

The resulting integral will resemble ∫ -2(-u)(e^u) du, leading to a manageable form for integration. This process shows how the chosen substitution links the variable x to its exponent, which is critical for exponentials,