Question

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.)

Answer 1

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