7.1 Integration By Parts/32

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

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

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

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

$={\frac {32}{5}}(\ln(2))^{2}-{\frac {64}{25}}(\ln(2))+{\frac {62}{125}}$

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