7.1 Integration By Parts/32

$f'(x)=\int _{0}^{t}e^{s}sin(t-s)\cdot ds$

$\int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-\int _{0}^{t}e^{s}\cos(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)-\int _{0}^{t}e^{s}\sin(t-s)\cdot ds$

$u=e^{s}~~~~~dv=\sin(t-s)dx~~~~~~~~~~~~~~~~u=e^{s}~~~~~dv=\cos(t-s)$

$du=e^{s}ds~~v=\cos(t-s)~~~~~~~~~~~~~~~~~~~du=e^{s}~~~~~v=-\sin(t-s)$

$\int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)-\int _{0}^{t}e^{s}\sin(t-s)\cdot ds$

$2\int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~e^{s}\cdot \cos(t-s)-e^{s}\cdot \sin(t-s)$

$\int _{0}^{t}e^{s}\sin(t-s)\cdot ds~~~=~~~{\frac {e^{s}}{2}}(\cos(t-s)-\sin(t-s)){\Bigg |}_{0}^{t}$

$={\frac {e^{t}}{2}}(\cos(0)-\sin(0))-{\frac {e^{0}}{2}}(\cos(t)-\sin(t))$

$={\frac {e^{t}}{2}}-{\frac {1}{2}}(\cos(t)-\sin(t))$

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