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))

\int \frac{1}{x ( x + 2 ) ^{2}} d x

8 7 x^{2} + 3

x \to - 2 lim (3 x^{4} + 2 x^{2} - x - 2)

\frac{d}{d x} (12 + 5 sin (π x))

f (x) = \frac{( x - 2 ) ( 3 + 5 x )}{x ^{2} + 1}

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))