You should use the following substitution such that:

`t^2 = u => 2t dt = du => dt = (du)/2sqrtu`

You need to change the variable such that:

`int e^(t^2) dt = int e^u*(1/(2sqrtu)) du`

You should use integration by parts using the following formula, such that:

`intf'*g =fg - int...

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You should use the following substitution such that:

`t^2 = u => 2t dt = du => dt = (du)/2sqrtu`

You need to change the variable such that:

`int e^(t^2) dt = int e^u*(1/(2sqrtu)) du`

You should use integration by parts using the following formula, such that:

`intf'*g =fg - int fg'`

`f = e^u =>f' = e^u du`

`g' = (1/(2sqrtu)) du => g = sqrt u`

`int e^u*(1/(2sqrtu)) du = e^u sqrt u - int e^u*sqrt u du`

Substituting back `t^2` for u yields:

`int_(-3x)^(2x) e^(t^2) dt = e^(t^2)*t|_(-3x)^(2x) - int_(-3x)^(2x) e^(t^2)*t dt`

If you continue to solve `int_(-3x)^(2x) e^(t^2)*t dt ` using parts, since there are no other methods indicated in this case, you will enter in an infinite loop, hence, you will need to use the imaginary error function.

**Hence, evaluating the given integral yields `int_(-3x)^(2x) e^(t^2) dt = e^(t^2)*t|_(-3x)^(2x) - int_(-3x)^(2x) e^(t^2)*t dt.` **