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

The integral of e^{-st}(cos(t)) is (e^{-st}sin(t))/(1+s^2)-(se^{-st}cos(t))/(1+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 e^{-st}(cos(t))

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

\frac{e ^{- s t} sin ( t )}{1 + s ^{2}} - \frac{s e ^{- s t} cos ( t )}{1 + s ^{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{e ^{- s t} sin ( t )}{1 + s ^{2}} - \frac{s e ^{- s t} cos ( t )}{1 + s ^{2}}

Add a constant to the solution

= \frac{e ^{- s t} sin ( t )}{1 + s ^{2}} - \frac{s e ^{- s t} cos ( t )}{1 + s ^{2}} + C

\int (100 + t)^{4} d t

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

\int - \frac{t ^{2}}{t ^{2} + 4 t + 3} d t

\int \frac{3 x + 1}{x ^{2} + 2 x + 2} d x

\int \frac{2 x ^{3}}{4 + x ^{8}} d x

What is the integral of e^{-st}(cos(t)) ?
The integral of e^{-st}(cos(t)) is (e^{-st}sin(t))/(1+s^2)-(se^{-st}cos(t))/(1+s^2)+C