Evaluate ∫ (2x)e^(x^2) dx using substitution. What is the result?

To evaluate the integral ∫ (2x)e^(x^2) dx, a suitable substitution is to let ( u = x^2 ). This leads to ( du = 2x , dx ). Consequently, the integral can be rewritten in terms of ( u ):

[

\int (2x) e^{x^2} , dx = \int e^u , du

]

The integral of ( e^u ) is simply ( e^u + C ). Substituting back ( u = x^2 ) gives:

[

e^{x^2} + C

]

Hence, this matches choice A. This choice is confirmed as the final result of the integral. The chosen substitution effectively transformed the original integral into a standard form that is well-known, leading directly to the correct answer.

This realization highlights the power of substitution techniques in integration, allowing the transformation of complex integrals into simpler forms that are easier to evaluate. The connection between ( x ) and ( u ) is crucial for successfully applying this method.