What is the integral from 0 to t of e^xsin(t-x) ?

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

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integral from 0 to t of e^xsin(t-x)

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

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

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

\int_{0}^{t} e^{x} sin (t - x) d x = \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^xsin(t-x) ?
The integral from 0 to t of e^xsin(t-x) is 1/2 e^t-1/2 cos(t)-1/2 sin(t)