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

The integral of e^{-xy}sin(x) is -(e^{-yx}cos(x))/(1+y^2)-(ye^{-yx}sin(x))/(1+y^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of e^{-xy}sin(x)

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

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

Solution steps

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

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

What is the integral of e^{-xy}sin(x) ?
The integral of e^{-xy}sin(x) is -(e^{-yx}cos(x))/(1+y^2)-(ye^{-yx}sin(x))/(1+y^2)+C