So first, we must solve for the antiderivative of the integrand. We will do this using u-substitution. inte^4tdt u=4t and du=4dt which can be rewritten as 1/4du=dt so that we can take out the constant and perform the operation of 1/4inte^udu. We know that the antiderivative of e^u is just e^u which leaves us with 1/4e^4t.
The next step is to evaluate the antiderivative at the given bounds. So 1/4e^4t evaluated at t=-1 is 1/4e^-4. The tricky part is to evaluate 1/4e^4t at =-oo. When we plug it in, it becomes 1/4e^-oo. That negative exponent can be brought down to the denominator of the fraction we have and it looks like 1/(4e^oo)
The denominator becomes oo because an exponential that is infinitely multiplied by itself tends toward infinity and that infinity multiplied by 4 is another version of infinity so the whole fraction now looks like 1/oo and since the denominator is infinitely large, that makes the whole value infinitely small, so we assume it to be 0.
Now, we subtract the two values we calculated, which looks like 1/4e^-4 - 0 which just equals 1/4e^-4 and the negative exponent brings the exponential to the denominator, giving us an answer 1/(4e^4)