What is the integral of e^{-2x}cos(2x+3) ?

The integral of e^{-2x}cos(2x+3) is (e^{-2x}sin(2x+3))/4-(e^{-2x}cos(2x+3))/4+C

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integral of e^{-2x}cos(2x+3)

\int e^{- 2 x} cos (2 x + 3) d x

\frac{e ^{- 2 x} sin ( 2 x + 3 )}{4} - \frac{e ^{- 2 x} cos ( 2 x + 3 )}{4} + C

Solution steps

\int e^{- 2 x} cos (2 x + 3) d x

Apply Integration By Parts

: \frac{1}{2} e^{- 2 x} sin (2 x + 3) - \int - e^{- 2 x} sin (2 x + 3) d x

= \frac{1}{2} e^{- 2 x} sin (2 x + 3) - \int - e^{- 2 x} sin (2 x + 3) d x

Take the constant out:

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

= \frac{1}{2} e^{- 2 x} sin (2 x + 3) - (- \int e^{- 2 x} sin (2 x + 3) d x)

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} cos (2 x + 3) - \int e^{- 2 x} cos (2 x + 3) d x

= \frac{1}{2} e^{- 2 x} sin (2 x + 3) - (- (- \frac{1}{2} e^{- 2 x} cos (2 x + 3) - \int e^{- 2 x} cos (2 x + 3) d x))

Therefore

\int e^{- 2 x} cos (2 x + 3) d x = \frac{1}{2} e^{- 2 x} sin (2 x + 3) - (- (- \frac{1}{2} e^{- 2 x} cos (2 x + 3) - \int e^{- 2 x} cos (2 x + 3) d x))

Isolate

\int e^{- 2 x} cos (2 x + 3) d x

= \frac{e ^{- 2 x} sin ( 2 x + 3 )}{4} - \frac{e ^{- 2 x} cos ( 2 x + 3 )}{4}

Add a constant to the solution

= \frac{e ^{- 2 x} sin ( 2 x + 3 )}{4} - \frac{e ^{- 2 x} cos ( 2 x + 3 )}{4} + C

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What is the integral of e^{-2x}cos(2x+3) ?
The integral of e^{-2x}cos(2x+3) is (e^{-2x}sin(2x+3))/4-(e^{-2x}cos(2x+3))/4+C