Evaluating the bounds of integral

Evaluating the bounds of integral

Given that $$f(x)\leq
C\epsilon^{-1}\int^1_x(1+\epsilon^{-k}e^{-\alpha(1-t)/\epsilon})e^{-\alpha(t-x)/\epsilon}dt.$$
I want to show that $$f(x)\leq
C(1+\epsilon^{-(k+1)}e^{-\alpha(1-x)/\epsilon}).$$On evaluating the
integral I get $$f(x)\leq
C\left(\epsilon(1-e^{-\alpha(1-x)/\epsilon})/\alpha+(1-x)\epsilon^{-k}e^{-\alpha(1-x)/\epsilon}\right)$$How
do I proceed?