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

The integral of e^{-θ}cos(θ/2) is -2(-1/5 e^{-θ}sin(θ/2)+2/5 e^{-θ}cos(θ/2))+C

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integral of e^{-θ}cos(θ/2)

\int e^{- θ} cos (\frac{θ}{2}) d θ

- 2 (- \frac{1}{5} e^{- θ} sin (\frac{θ}{2}) + \frac{2}{5} e^{- θ} cos (\frac{θ}{2})) + C

Solution steps

\int e^{- θ} cos (\frac{θ}{2}) d θ

Apply u-substitution

: \int - e^{u} cos (\frac{u}{2}) d u

= \int - e^{u} cos (\frac{u}{2}) d u

Take the constant out:

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

= - \int e^{u} cos (\frac{u}{2}) d u

Apply u-substitution

: \int e^{2 v} cos (v) \cdot 2 d v

= - \int e^{2 v} cos (v) \cdot 2 d v

Take the constant out:

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

= - 2 \cdot \int e^{2 v} cos (v) d v

Apply Integration By Parts

: e^{2 v} sin (v) - \int e^{2 v} \cdot 2 sin (v) d v

= - 2 (e^{2 v} sin (v) - \int e^{2 v} \cdot 2 sin (v) d v)

Take the constant out:

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

= - 2 (e^{2 v} sin (v) - 2 \cdot \int e^{2 v} sin (v) d v)

Apply Integration By Parts

: - e^{2 v} cos (v) - \int - 2 e^{2 v} cos (v) d v

= - 2 (e^{2 v} sin (v) - 2 (- e^{2 v} cos (v) - \int - 2 e^{2 v} cos (v) d v))

Take the constant out:

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

= - 2 (e^{2 v} sin (v) - 2 (- e^{2 v} cos (v) - (- 2 \cdot \int e^{2 v} cos (v) d v)))

Therefore

- 2 \cdot \int e^{2 v} cos (v) d v = - 2 (e^{2 v} sin (v) - 2 (- e^{2 v} cos (v) - (- 2 \cdot \int e^{2 v} cos (v) d v)))

Isolate

\int e^{2 v} cos (v) d v

= - 2 (\frac{e ^{2 v} sin ( v )}{5} + \frac{2 e ^{2 v} cos ( v )}{5})

Substitute back

= - 2 (\frac{e ^{2 \cdot \frac{- θ}{2}} sin ( \frac{- θ}{2} )}{5} + \frac{2 e ^{2 \cdot \frac{- θ}{2}} cos ( \frac{- θ}{2} )}{5})

Simplify

- 2 (\frac{e ^{2 \cdot \frac{- θ}{2}} sin ( \frac{- θ}{2} )}{5} + \frac{2 e ^{2 \cdot \frac{- θ}{2}} cos ( \frac{- θ}{2} )}{5}) : - 2 (- \frac{1}{5} e^{- θ} sin (\frac{θ}{2}) + \frac{2}{5} e^{- θ} cos (\frac{θ}{2}))

= - 2 (- \frac{1}{5} e^{- θ} sin (\frac{θ}{2}) + \frac{2}{5} e^{- θ} cos (\frac{θ}{2}))

Add a constant to the solution

= - 2 (- \frac{1}{5} e^{- θ} sin (\frac{θ}{2}) + \frac{2}{5} e^{- θ} cos (\frac{θ}{2})) + C

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What is the integral of e^{-θ}cos(θ/2) ?
The integral of e^{-θ}cos(θ/2) is -2(-1/5 e^{-θ}sin(θ/2)+2/5 e^{-θ}cos(θ/2))+C