f ′ ( x ) = ∫ 0 t e s s i n ( t − s ) ⋅ d s {\displaystyle f'(x)=\int _{0}^{t}e^{s}sin(t-s)\cdot ds}
∫ 0 t e s sin ( t − s ) ⋅ d s = e s ⋅ cos ( t − s ) − ∫ 0 t e s cos ( t − s ) ⋅ d s = e s ⋅ cos ( t − s ) − e s ⋅ sin ( t − s ) − ∫ 0 t e s sin ( t − s ) ⋅ d s {\displaystyle \int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-\int _{0}^{t}e^{s}\cos(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)-\int _{0}^{t}e^{s}\sin(t-s)\cdot ds}
u = e s d v = sin ( t − s ) d x u = e s d v = cos ( t − s ) {\displaystyle u=e^{s}~~~~~dv=\sin(t-s)dx~~~~~~~~~~~~~~~~u=e^{s}~~~~~dv=\cos(t-s)}
d u = e s d s v = cos ( t − s ) d u = e s v = − sin ( t − s ) {\displaystyle du=e^{s}ds~~v=\cos(t-s)~~~~~~~~~~~~~~~~~~~du=e^{s}~~~~~v=-\sin(t-s)}
∫ 0 t e s sin ( t − s ) ⋅ d s = e s ⋅ cos ( t − s ) − e s ⋅ sin ( t − s ) − ∫ 0 t e s sin ( t − s ) ⋅ d s {\displaystyle \int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)-\int _{0}^{t}e^{s}\sin(t-s)\cdot ds}
2 ∫ 0 t e s sin ( t − s ) ⋅ d s = e s ⋅ cos ( t − s ) − e s ⋅ sin ( t − s ) {\displaystyle 2\int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)}
∫ 0 t e s sin ( t − s ) ⋅ d s = e s 2 ( cos ( t − s ) − sin ( t − s ) ) | 0 t {\displaystyle \int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~{\frac {e^{s}}{2}}(\cos(t-s)-\sin(t-s)){\Bigg |}_{0}^{t}}
= e t 2 ( cos ( 0 ) − sin ( 0 ) ) − e 0 2 ( cos ( t ) − sin ( t ) ) {\displaystyle ={\frac {e^{t}}{2}}(\cos(0)-\sin(0))-{\frac {e^{0}}{2}}(\cos(t)-\sin(t))}
= e t 2 − 1 2 ( cos ( t ) − sin ( t ) ) {\displaystyle ={\frac {e^{t}}{2}}-{\frac {1}{2}}(\cos(t)-\sin(t))}
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