What is the integral of cos(kt)e^{-st} ?

The integral of cos(kt)e^{-st} is (ke^{-st}sin(kt))/(k^2+s^2)-(se^{-st}cos(kt))/(k^2+s^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos(kt)e^{-st}

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

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

Solution steps

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

Apply Integration By Parts

: e^{- s t} \frac{1}{k} sin (k t) - \int - s e^{- s t} \frac{1}{k} sin (k t) d t

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

Take the constant out:

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

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

Simplify

- s \frac{1}{k} : - \frac{s}{k}

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

Apply Integration By Parts

: - e^{- s t} \frac{1}{k} cos (k t) - \int s e^{- s t} \frac{1}{k} cos (k t) d t

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

Take the constant out:

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

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

Simplify

s \frac{1}{k} : \frac{s}{k}

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int k x d x

\int (x - 1)^{5} + 3 (x - 1)^{2} + 5 d x

\int (x + 5) (x - 2) d x

\int - \frac{1800}{x ^{2}} d x

\int \frac{1}{x ln ^{n} ( x )} d x

What is the integral of cos(kt)e^{-st} ?
The integral of cos(kt)e^{-st} is (ke^{-st}sin(kt))/(k^2+s^2)-(se^{-st}cos(kt))/(k^2+s^2)+C