What is the integral from 0 to t of e^ssin(t-s) ?

The integral from 0 to t of e^ssin(t-s) is 1/2 e^t-1/2 cos(t)-1/2 sin(t)

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integral from 0 to t of e^ssin(t-s)

\int_{0}^{t} e^{s} sin (t - s) d s

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

Solution steps

\int_{0}^{t} e^{s} sin (t - s) d s

Compute the indefinite integral:

\int e^{s} sin (t - s) d s = \frac{e ^{s} cos ( t - s )}{2} + \frac{e ^{s} sin ( t - s )}{2} + C

Compute the boundaries:

\int_{0}^{t} e^{s} sin (t - s) d s = \frac{1}{2} e^{t} - (\frac{1}{2} cos (t) + \frac{1}{2} sin (t))

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

Simplify

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

What is the integral from 0 to t of e^ssin(t-s) ?
The integral from 0 to t of e^ssin(t-s) is 1/2 e^t-1/2 cos(t)-1/2 sin(t)