81. Possible and impossible integrals
Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.
a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.

Verified step by step guidance

1. Start with the integral definition: \(I_n = \int x^n e^{-x^2} \, dx\). For \(n=1\), we have \(I_1 = \int x e^{-x^2} \, dx\).
2. Recognize that the integrand \(x e^{-x^2}\) suggests a substitution because the derivative of \(-x^2\) is \(-2x\), which is closely related to the \(x\) term in the integrand.
3. Use the substitution method: let \(u = -x^2\), then compute \(du = -2x \, dx\), which implies \(x \, dx = -\frac{1}{2} du\).
4. Rewrite the integral in terms of \(u\): \(I_1 = \int x e^{-x^2} \, dx = \int e^u \left(-\frac{1}{2} du\right) = -\frac{1}{2} \int e^u \, du\).
5. Integrate with respect to \(u\): \(-\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C\). Finally, substitute back \(u = -x^2\) to express the answer in terms of \(x\).

Key Concepts

Integration of Functions Involving Exponentials and Polynomials

This concept involves integrating products of polynomial terms and exponential functions, such as xⁿ e⁻ˣ². Techniques like substitution and integration by parts are often used to simplify these integrals, especially when direct antiderivatives are not straightforward.

Integration by Parts

Integration by parts is a method based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often reducing the power of polynomials or simplifying exponential terms, which is essential for evaluating integrals like I₁ = ∫ x e⁻ˣ² dx.

Non-Elementary Integrals and Special Functions

Some integrals, such as ∫ e⁻ˣ² dx, cannot be expressed in terms of elementary functions and are instead represented using special functions like the error function (erf). Recognizing when an integral falls into this category helps in understanding the limitations of standard integration techniques.