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

The integral of e^{-3x}cos(x) is (e^{-3x}sin(x))/(10)-(3e^{-3x}cos(x))/(10)+C

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

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

\frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10} + C

Solution steps

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

Apply Integration By Parts

: e^{- 3 x} sin (x) - \int - 3 e^{- 3 x} sin (x) d x

= e^{- 3 x} sin (x) - \int - 3 e^{- 3 x} sin (x) d x

Take the constant out:

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

= e^{- 3 x} sin (x) - (- 3 \cdot \int e^{- 3 x} sin (x) d x)

Apply Integration By Parts

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

= e^{- 3 x} sin (x) - (- 3 (- e^{- 3 x} cos (x) - \int 3 e^{- 3 x} cos (x) d x))

Take the constant out:

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

= e^{- 3 x} sin (x) - (- 3 (- e^{- 3 x} cos (x) - 3 \cdot \int e^{- 3 x} cos (x) d x))

Therefore

\int e^{- 3 x} cos (x) d x = e^{- 3 x} sin (x) - (- 3 (- e^{- 3 x} cos (x) - 3 \cdot \int e^{- 3 x} cos (x) d x))

Isolate

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

= \frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10}

Add a constant to the solution

= \frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10} + C

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\frac{d}{d x} (\frac{x ^{4}}{8} + \frac{1}{4 x ^{2}})

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What is the integral of e^{-3x}cos(x) ?
The integral of e^{-3x}cos(x) is (e^{-3x}sin(x))/(10)-(3e^{-3x}cos(x))/(10)+C