What is the integral from 0 to t of-5e^ssin(t-s) ?

The integral from 0 to t of-5e^ssin(t-s) is -5/2 e^t+5(1/2 cos(t)+1/2 sin(t))

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integral from 0 to t of-5e^ssin(t-s)

\int_{0}^{t} - 5 e^{s} sin (t - s) d s

- \frac{5}{2} e^{t} + 5 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t))

Solution steps

\int_{0}^{t} - 5 e^{s} sin (t - s) d s

Compute the indefinite integral:

\int - 5 e^{s} sin (t - s) d s = - 5 (\frac{e ^{s} cos ( - s + t )}{2} + \frac{e ^{s} sin ( - s + t )}{2}) + C

Compute the boundaries:

\int_{0}^{t} - 5 e^{s} sin (t - s) d s = - \frac{5}{2} e^{t} - (- 5 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t)))

= - \frac{5}{2} e^{t} - (- 5 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t)))

Simplify

= - \frac{5}{2} e^{t} + 5 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t))

\int (x + 2) (x - 7) d x

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\frac{d}{d x} (a e^{4 x})

\int \frac{2 x ^{2} - 6}{x ^{3} + x} d x

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What is the integral from 0 to t of-5e^ssin(t-s) ?
The integral from 0 to t of-5e^ssin(t-s) is -5/2 e^t+5(1/2 cos(t)+1/2 sin(t))