What is the integral of e^tsin(t) ?

The integral of e^tsin(t) is -(e^tcos(t))/2+(e^tsin(t))/2+C

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integral of e^tsin(t)

\int e^{t} sin (t) d t

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

Solution steps

\int e^{t} sin (t) d t

Apply Integration By Parts

: - e^{t} cos (t) - \int - e^{t} cos (t) d t

= - e^{t} cos (t) - \int - e^{t} cos (t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{t} cos (t) - (- \int e^{t} cos (t) d t)

Apply Integration By Parts

: e^{t} sin (t) - \int e^{t} sin (t) d t

= - e^{t} cos (t) - (- (e^{t} sin (t) - \int e^{t} sin (t) d t))

Therefore

\int e^{t} sin (t) d t = - e^{t} cos (t) - (- (e^{t} sin (t) - \int e^{t} sin (t) d t))

Isolate

\int e^{t} sin (t) d t

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

Add a constant to the solution

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

y^{^{'}} = 1 + 4 y^{2}

\frac{\partial}{\partial t} (e^{2 t})

f (x) = \frac{6 x}{x + 5}, at x = - 3

\frac{d}{d x} (ln (2 x^{2} + 5))

What is the integral of e^tsin(t) ?
The integral of e^tsin(t) is -(e^tcos(t))/2+(e^tsin(t))/2+C