Solution: This problem needs to be broken up into steps.

Step 1: Rewrite by breaking up the problem into two integrals:

ò (4*x*^{2} + 1)*dx *= ò 4*x*^{2} *dx* + ò 1 *dx *(Sum Rule)

Step 2: Use the Constant Multiple Rule and place a *x*^{0} into the second integral:

ò 4*x*^{2} *dx* + ò 1 *dx *= 4 ò *x*^{2} *dx* + 1 ò *x*^{0} *dx*

Step 3: Finally, use the Power Rule on both of the integrals and combine the constants into one:

4 ò *x*^{2} *dx* + 1 ò *x*^{0} *dx* =
*x*^{(2 + 1) }+ *C* +
*x*^{(0 + 1)} + *C* (Power Rule)

=
*x*^{3} + *x* + *C*,

where C is an arbitrary constant.

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