What is the integral from 0 to t of e^vsin(t-v) ?

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

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integral from 0 to t of e^vsin(t-v)

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

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

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

\int_{- 1}^{1} \frac{cos ( x )}{1 + x ^{4}} d x

\int_{0}^{2} \frac{x ^{3}}{3} + x^{2} - 3 x d x

\int_{0}^{π} 3 d x

\int_{1}^{0} ln (x) d x

\int_{1}^{4} \frac{1}{( x - 2 ) ^{\frac{3}{2}}} d x

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