A life annuity on $(80)$ provides for benefits made continuously for $2$ years, at the annual rate of $c(t)$ at time $t$, provided that $(x)$ is alive. Suppose that $c(t)=t$, $0 \le t \le 2$,and the interest rate is $0$. Find the present value under each of the following assumptions.
(a) $q_{80}=0.09$, $q_{81}=0.12$, and UDD holds.
(b) The force of mortality is given by
$\mu_{80}(t)=\frac{1}{10-t}$, $0 \le t \le 10$
$a_x(c)=\int ^2_0 v(t)c(t) _tP_x$ $dt$
$P_x=\frac{(1-t)l_{80}+tl_{81}}{l_{80}}=1-0.09t$
$_2P_x=\frac{(2-t)l_{81}+(t-1)l_{82}}{l_{80}}=0.03+0.88t$
$v(t)=1$
$a_x(c)=\int^1_0(t-0.09t^2)$ $dt$ $+\int^2_1(0.03t+0.88t^2)$ $dt=2.56$
Answer is $1.7444$ and Ido not know the second part.
Your calculation of ${}_2 p_x = 0.03 + 0.88t$ is clearly wrong; if $t = 2$, this value exceeds $1$. This is the kind of error you need to be able to spot before you even attempt to proceed to integrate, otherwise you are just wasting your time. You also need to be precise about your notation: here, $x = 80$ unless otherwise indicated; and the expression ${}_2 p_x$ is not a function of $t$.
You need to write either a piecewise definition of ${}_t p_{80}$, the survival probability of $(80)$ to time $t$, or write ${}_t p_{80}$ and ${}_t p_{81}$ separately for $t \in [0,1]$ under the UDD assumption, and note that for $t \in (1,2]$, we have $${}_t p_{80} = (p_{80})({}_{t-1} p_{81}).$$ It is this latter approach that I recommend.