7.1 Integration By Parts/32

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

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

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

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

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

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

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

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

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

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