What is the integral of e^{-5x}cos(x) ?

The integral of e^{-5x}cos(x) is (e^{-5x}sin(x))/(26)-(5e^{-5x}cos(x))/(26)+C

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integral of e^{-5x}cos(x)

\int e^{- 5 x} cos (x) d x

\frac{e ^{- 5 x} sin ( x )}{26} - \frac{5 e ^{- 5 x} cos ( x )}{26} + C

Solution steps

\int e^{- 5 x} cos (x) d x

Apply Integration By Parts

: e^{- 5 x} sin (x) - \int - 5 e^{- 5 x} sin (x) d x

= e^{- 5 x} sin (x) - \int - 5 e^{- 5 x} sin (x) d x

Take the constant out:

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

= e^{- 5 x} sin (x) - (- 5 \cdot \int e^{- 5 x} sin (x) d x)

Apply Integration By Parts

: - e^{- 5 x} cos (x) - \int 5 e^{- 5 x} cos (x) d x

= e^{- 5 x} sin (x) - (- 5 (- e^{- 5 x} cos (x) - \int 5 e^{- 5 x} cos (x) d x))

Take the constant out:

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

= e^{- 5 x} sin (x) - (- 5 (- e^{- 5 x} cos (x) - 5 \cdot \int e^{- 5 x} cos (x) d x))

Therefore

\int e^{- 5 x} cos (x) d x = e^{- 5 x} sin (x) - (- 5 (- e^{- 5 x} cos (x) - 5 \cdot \int e^{- 5 x} cos (x) d x))

Isolate

\int e^{- 5 x} cos (x) d x

= \frac{e ^{- 5 x} sin ( x )}{26} - \frac{5 e ^{- 5 x} cos ( x )}{26}

Add a constant to the solution

= \frac{e ^{- 5 x} sin ( x )}{26} - \frac{5 e ^{- 5 x} cos ( x )}{26} + C

\frac{\partial}{\partial y} (3 x^{2} + 2 y^{2})

(y + 1) d x + (y - 1) (1 + x^{2}) d y = 0

\frac{\partial}{\partial y} (- 3 y^{4} sin (3 x))

\frac{d}{d x} (4 x^{2} + 5 x - 10)

x^{^{''}} + 9 x^{^{'}} = 27, x (0) = 4, x^{^{'}} (0) = 6

What is the integral of e^{-5x}cos(x) ?
The integral of e^{-5x}cos(x) is (e^{-5x}sin(x))/(26)-(5e^{-5x}cos(x))/(26)+C