What is the integral of cos(5t)e^t ?

The integral of cos(5t)e^t is (5e^tsin(5t))/(26)+(e^tcos(5t))/(26)+C

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integral of cos(5t)e^t

\int cos (5 t) e^{t} d t

\frac{5 e ^{t} sin ( 5 t )}{26} + \frac{e ^{t} cos ( 5 t )}{26} + C

Solution steps

\int cos (5 t) e^{t} d t

Apply Integration By Parts

= e^{t} \frac{1}{5} sin (5 t) - \int e^{t} \frac{1}{5} sin (5 t) d t

Take the constant out:

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

= e^{t} \frac{1}{5} sin (5 t) - \frac{1}{5} \cdot \int e^{t} sin (5 t) d t

Apply Integration By Parts

= e^{t} \frac{1}{5} sin (5 t) - \frac{1}{5} (- \frac{1}{5} e^{t} cos (5 t) - \int - \frac{1}{5} e^{t} cos (5 t) d t)

Take the constant out:

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

= e^{t} \frac{1}{5} sin (5 t) - \frac{1}{5} (- \frac{1}{5} e^{t} cos (5 t) - (- \frac{1}{5} \cdot \int e^{t} cos (5 t) d t))

Therefore

\int e^{t} cos (5 t) d t = e^{t} \frac{1}{5} sin (5 t) - \frac{1}{5} (- \frac{1}{5} e^{t} cos (5 t) - (- \frac{1}{5} \cdot \int e^{t} cos (5 t) d t))

Isolate

\int e^{t} cos (5 t) d t

= \frac{5 e ^{t} sin ( 5 t )}{26} + \frac{e ^{t} cos ( 5 t )}{26}

Add a constant to the solution

= \frac{5 e ^{t} sin ( 5 t )}{26} + \frac{e ^{t} cos ( 5 t )}{26} + C

What is the integral of cos(5t)e^t ?
The integral of cos(5t)e^t is (5e^tsin(5t))/(26)+(e^tcos(5t))/(26)+C