Alright, let's work through this integral step-by-step together.
We are given the following integral:
[tex]\[ \int_x^{80} e^{2 - t} \, dt \][/tex]
Here, [tex]\( t \)[/tex] is the variable of integration, and the limits of integration are from [tex]\( x \)[/tex] to 80. Let's proceed by solving this definite integral.
1. Determine the antiderivative:
The integrand [tex]\( e^{2 - t} \)[/tex] can be integrated using the standard rules for integration of exponential functions. We start by noting that the integral of [tex]\( e^{kt} \)[/tex] is [tex]\(\frac{1}{k} e^{kt} \)[/tex] for some constant [tex]\( k \)[/tex]. To utilize this, let's perform a substitution.
Let:
[tex]\[
u = 2 - t
\][/tex]
Then, the differential [tex]\( du = -dt \)[/tex], or [tex]\( dt = -du \)[/tex].
2. Substitute and integrate:
Substituting [tex]\( u = 2 - t \)[/tex], the integral becomes:
[tex]\[
\int_x^{80} e^{2 - t} \, dt = \int_{x}^{80} e^{u} \cdot (-du)
\][/tex]
Removing the negative sign and adjusting the limits for the substitution, we get:
[tex]\[
= -\int_{2 - x}^{2 - 80} e^{u} \, du
\][/tex]
The limits of integration are now adjusted to [tex]\( 2 - x \)[/tex] to [tex]\( 2 - 80 \)[/tex].
Simplifying the limits:
[tex]\[
-\int_{2 - x}^{-78} e^{u} \, du
\][/tex]
3. Resolve the integral:
The integral of [tex]\( e^u \)[/tex] is just [tex]\( e^u \)[/tex]. Hence, integrating from [tex]\( 2 - x \)[/tex] to [tex]\( -78 \)[/tex] and applying the fundamental theorem of calculus, we have:
[tex]\[
- \left[ e^u \right]_{2 - x}^{-78}
\][/tex]
Which simplifies to:
[tex]\[
- \left( e^{-78} - e^{2 - x} \right)
\][/tex]
4. Simplify the expression:
Expanding the negative sign through the parentheses gives:
[tex]\[
= -e^{-78} + e^{2 - x}
\][/tex]
Therefore, the final result of the integral is:
[tex]\[
e^{2 - x} - e^{-78}
\][/tex]
In summary, the evaluated result of the integral is:
[tex]\[
\boxed{e^{2 - x} - e^{-78}}
\][/tex]
