Integrate the function xsin(4x^2) with respect to x, using the integration by substitution method.

1. To solve the integral of the function xsin(4x^2) with respect to the variable x, one can employ the technique of integration by substitution. This method is particularly useful when dealing with composite functions, as it allows for a simplification of the integral by transforming it into a more manageable form. In this case, we can identify a suitable substitution that will facilitate the integration process.

2. A logical choice for substitution in this scenario is to let u equal 4x^2. Consequently, the differential du can be expressed as 8x dx, which implies that dx can be rewritten in terms of du. This transformation will enable us to express the original integral in terms of u, thereby simplifying the integration process. Specifically, we can rearrange the expression to isolate dx, yielding dx = du/(8x). Substituting these expressions into the integral will allow us to eliminate the variable x from the integral.

3. After performing the substitution, the integral can be rewritten in terms of u, leading to a new integral that is easier to evaluate. The resulting expression will involve the sine function of u, which can be integrated using standard techniques. Once the integration is completed, it is essential to revert back to the original variable x by substituting u back with 4x^2. This final step will yield the solution to the integral of the original function xsin(4x^2) with respect to x, completing the process of integration by substitution.