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

The integral of e^{5t}cos(t) is (e^{5t}sin(t))/(26)+(5e^{5t}cos(t))/(26)+C

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integral of e^{5t}cos(t)

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

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

Solution steps

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

Apply Integration By Parts

: e^{5 t} sin (t) - \int e^{5 t} \cdot 5 sin (t) d t

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int \frac{1}{4 - 49 x ^{2}} d x

y = e^{- k x^{2}}

1 + x^{2}

\frac{d}{d x} (e^{x^{5} + 3 x})

x^{3} - 8 x^{2}

What is the integral of e^{5t}cos(t) ?
The integral of e^{5t}cos(t) is (e^{5t}sin(t))/(26)+(5e^{5t}cos(t))/(26)+C