What is the integral of e^{-2*pi*x}*cos(pi*x) ?

The integral of e^{-2*pi*x}*cos(pi*x) is 1/(5pi)(e^{-2pix}sin(pix)-2e^{-2pix}cos(pix))+C

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integral of e^{-2pix}cos(pix)

\int e^{- 2 \cdot π \cdot x} \cdot cos (π \cdot x) d x

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

Solution steps

\int e^{- 2 π x} cos (π x) d x

Apply u-substitution

: \int e^{- 2 u} cos (u) \frac{1}{π} d u

= \int e^{- 2 u} cos (u) \frac{1}{π} d u

Take the constant out:

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

= \frac{1}{π} \cdot \int e^{- 2 u} cos (u) d u

Apply Integration By Parts

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

= \frac{1}{π} (e^{- 2 u} sin (u) - \int - 2 e^{- 2 u} sin (u) d u)

Take the constant out:

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

= \frac{1}{π} (e^{- 2 u} sin (u) - (- 2 \cdot \int e^{- 2 u} sin (u) d u))

Apply Integration By Parts

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

= \frac{1}{π} (e^{- 2 u} sin (u) - (- 2 (- e^{- 2 u} cos (u) - \int 2 e^{- 2 u} cos (u) d u)))

Take the constant out:

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

= \frac{1}{π} (e^{- 2 u} sin (u) - (- 2 (- e^{- 2 u} cos (u) - 2 \cdot \int e^{- 2 u} cos (u) d u)))

Therefore

\frac{1}{π} \cdot \int e^{- 2 u} cos (u) d u = \frac{1}{π} (e^{- 2 u} sin (u) - (- 2 (- e^{- 2 u} cos (u) - 2 \cdot \int e^{- 2 u} cos (u) d u)))

Isolate

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

= \frac{1}{π} (\frac{e ^{- 2 u} sin ( u )}{5} - \frac{2 e ^{- 2 u} cos ( u )}{5})

Substitute back

u = π x

= \frac{1}{π} (\frac{e ^{- 2 π x} sin ( π x )}{5} - \frac{2 e ^{- 2 π x} cos ( π x )}{5})

Simplify

\frac{1}{π} (\frac{e ^{- 2 π x} sin ( π x )}{5} - \frac{2 e ^{- 2 π x} cos ( π x )}{5}) : \frac{1}{5 π} (e^{- 2 π x} sin (π x) - 2 e^{- 2 π x} cos (π x))

= \frac{1}{5 π} (e^{- 2 π x} sin (π x) - 2 e^{- 2 π x} cos (π x))

Add a constant to the solution

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

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