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)

x \to 0 + lim (ln (2 x))

\frac{d y}{d t} + y = 5 sin (2 t)

\frac{d}{d x} (- 3 x + 1)

\frac{( 1 + e ^{x} )}{1 - e ^{x}}

x \to - 2 lim ((3 x - 8)^{2})

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)