What is the integral of e^{-sx}sin(x) ?

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\frac{d}{d x} (4 x^{3} + 3 x^{2} + 2 x)

\frac{d}{d x} (x^{3} - 100 x + 1)

\frac{d ^{2}}{d x ^{2}} (e^{7 x - 4})

\int arctan (2 t) d t

f (x) = x^{4} - 8 x^{3}

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