What is the integral of e^t*cos(t) ?

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

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integral of e^t*cos(t)

\int e^{t} \cdot cos (t) d t

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

Solution steps

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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