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

The integral of e^{-θ}cos(θ) is (e^{-θ}sin(θ))/2-(e^{-θ}cos(θ))/2+C

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integral of e^{-θ}cos(θ)

\int e^{- θ} cos (θ) d θ

\frac{e ^{- θ} sin ( θ )}{2} - \frac{e ^{- θ} cos ( θ )}{2} + C

Solution steps

\int e^{- θ} cos (θ) d θ

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

\int e^{- θ} cos (θ) d θ

= \frac{e ^{- θ} sin ( θ )}{2} - \frac{e ^{- θ} cos ( θ )}{2}

Add a constant to the solution

= \frac{e ^{- θ} sin ( θ )}{2} - \frac{e ^{- θ} cos ( θ )}{2} + C

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What is the integral of e^{-θ}cos(θ) ?
The integral of e^{-θ}cos(θ) is (e^{-θ}sin(θ))/2-(e^{-θ}cos(θ))/2+C