Questions: Evaluate the integral. [ int0^infty 2 x e^x^2 d x ]

Solution Steps

To evaluate the given integral, we need to analyze the behavior of the integrand \( 2x e^{x^2} \) as \( x \) approaches infinity. We can use substitution to simplify the integral and determine if it converges or diverges.

We are given the integral: \[ \int_{0}^{\infty} 2 x e^{x^{2}} \, dx \]

The integrand is \( 2x e^{x^2} \). As \( x \) approaches infinity, \( e^{x^2} \) grows extremely rapidly.

To evaluate the integral, we can use substitution. Let \( u = x^2 \), then \( du = 2x \, dx \). The integral becomes: \[ \int_{0}^{\infty} e^{u} \, du \]

The integral \( \int_{0}^{\infty} e^{u} \, du \) diverges because \( e^{u} \) grows exponentially as \( u \) approaches infinity.

Final Answer