What is the integral from 0 to t of-8e^ssin(t-s) ?

The integral from 0 to t of-8e^ssin(t-s) is -4e^t+8(1/2 cos(t)+1/2 sin(t))

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integral from 0 to t of-8e^ssin(t-s)

\int_{0}^{t} - 8 e^{s} sin (t - s) d s

- 4 e^{t} + 8 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t))

Solution steps

\int_{0}^{t} - 8 e^{s} sin (t - s) d s

Compute the indefinite integral:

\int - 8 e^{s} sin (t - s) d s = - 8 (\frac{e ^{s} cos ( - s + t )}{2} + \frac{e ^{s} sin ( - s + t )}{2}) + C

Compute the boundaries:

\int_{0}^{t} - 8 e^{s} sin (t - s) d s = - 4 e^{t} - (- 8 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t)))

= - 4 e^{t} - (- 8 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t)))

Simplify

= - 4 e^{t} + 8 (\frac{1}{2} cos (t) + \frac{1}{2} sin (t))

\int \frac{x - 3}{x ^{2} - 6 x - 1} d x

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\frac{d}{d x} (\frac{2 x}{x + 3})

n = 8 \sum \infty e^{3 - 4 n}

\int \frac{1}{2 x ^{3}} d x

What is the integral from 0 to t of-8e^ssin(t-s) ?
The integral from 0 to t of-8e^ssin(t-s) is -4e^t+8(1/2 cos(t)+1/2 sin(t))