Which substitution might be useful for integrating ∫ x*e^(x^2) dx?

To solve the integral ∫ x*e^(x^2) dx, the substitution u = x^2 is particularly effective because it simplifies the integrand significantly. By using this substitution, we can express the differential dx in terms of du.

When we make the substitution u = x^2, we differentiate both sides to find:

du/dx = 2x

This gives us:

du = 2x dx

or

dx = du / (2x).

Now we can rewrite the integral in terms of u. The expression for x can be replaced with √u, but since we have x dx from the original integral, we can directly substitute:

∫ x * e^(x^2) dx = ∫ x * e^u * (du / (2x)) = ∫ (1/2) e^u du.

This substitution effectively simplifies the integral to a form that is straightforward to compute.

Using another substitution, such as u = e^(x^2) or u = ln(x), does not lead to a similar simplification. They would involve more complicated relationships that do not directly relate to the structure of the integral provided. Thus, using u = x^2 streamlines the integration