Evaluate the integral.
1/2 integral e^(t/2)
(I'm not sure what the 1/2 on the left of the integral symbol means.)
A constant factor within an integrand can be separated from the integrand and multiplied by the integral.
In this case:
integral [ ( 1 / 2 ) * e ^ ( t / 2 ) dt ]
= 1 / 2 integral e ^ ( t / 2 )
Substitute:
u = t / 2
du = dt / 2 Multiply both sides by 2
2 du = dt
dt = 2 du
1 / 2 integral e ^ ( t / 2 ) dt =
1 / 2 integral e ^ ( u ) 2 * du =
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Again a constant factor, in this case 2 multiplied by the integral.
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( 1 / 2 ) * 2 * integral e ^ ( u ) du =
integral e ^ ( u ) du =
e ^ ( u ) + C =
e ^ ( t / 2 ) + C
1 / 2 integral e ^ ( t / 2 ) dt = e ^ ( t / 2 ) + C