What is the integral of e^{-ax}cos(x) ?

The integral of e^{-ax}cos(x) is (e^{-ax}sin(x))/(1+a^2)-(ae^{-ax}cos(x))/(1+a^2)+C

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integral of e^{-ax}cos(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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What is the integral of e^{-ax}cos(x) ?
The integral of e^{-ax}cos(x) is (e^{-ax}sin(x))/(1+a^2)-(ae^{-ax}cos(x))/(1+a^2)+C