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 s i n ( t − s ) ⋅ d s = e s ⋅ c o s ( t − s ) − ∫ 0 t e s c o s ( t − s ) ⋅ d s = e s ⋅ c o s ( t − s ) − e s ⋅ s i n ( 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 = s i n ( t − s ) d x u = e s d v = c o s ( 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 = c o s ( t − s ) d u = e s v = − s i n ( t − s ) {\displaystyle du=e^{s}ds~~v=cos(t-s)~~~~~~~~~~~~~~~~~~~du=e^{s}~~~~~v=-sin(t-s)}
∫ 0 t e s s i n ( t − s ) ⋅ d s = e s ⋅ c o s ( t − s ) − e s ⋅ s i n ( 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 s i n ( t − s ) ⋅ d s = e s ⋅ c o s ( t − s ) − e s ⋅ s i n ( 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 s i n ( t − s ) ⋅ d s = e s 2 ( c o s ( t − s ) − s i n ( 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 s 2 ( c o s ( 0 ) − s i n ( 0 ) ) − e s 2 ( c o s ( t ) − s i n ( t ) ) {\displaystyle ={\frac {e^{s}}{2}}(cos(0)-sin(0))-{\frac {e^{s}}{2}}(cos(t)-sin(t))}
1 2 3 4 5 7 8 9 10 11 12 13 14 15 17 18 19 20 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 43 45 47 48 49 50 51 52 53 54 61 65
