Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 36

You need to use integration by parts, such that:
int udv = uv - int vdu
u = e^s => du = e^s ds
dv = sin(t-s) => v = (-cos(t-s))/(-1)
int e^s sin (t-s) ds = e^s*cos(t-s) - int e^s*cos(t-s)ds
You need to use parts again, for int e^s*cos(t-s)ds , such that:
u = e^s => du = e^s ds
dv = cos(t-s) => v = (sin(t-s))/(-1)
int e^s*cos(t-s)ds = -e^s*sin (t-s) + int e^s*sin (t-s) ds
Replacing back, yields:
int e^s sin (t-s) ds = e^s*cos(t-s) - (-e^s*sin (t-s) + int e^s*sin (t-s) ds)
You need to use the substitution int e^s sin (t-s) ds= I:
I = e^s*cos(t-s) + e^s*sin (t-s)- I
2I = e^s*cos(t-s) + e^s*sin (t-s) => I = (e^s*(cos(t-s)+sin(t-s)))/2
You need to evaluate the definite integral, using the fundamental theorem of calculus, such that:

int_0^t e^s sin (t-s) ds = (e^s*(cos(t-s)+sin(t-s)))/2|_0^t
int_0^t e^s sin (t-s) ds = (e^t*(cos(t-t)+sin(t-t)) - e^0*(cos(t-0)+sin(t-0)))/2

int_0^t e^s sin (t-s) ds = (e^t - cos t - sin t)/2
Hence, evaluating the definite integral, using integration by parts, yields int_0^t e^s sin (t-s) ds = (e^t - cos t - sin t)/2.