What is the integral of (cos(5x))/(e^x) ?

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

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integral of (cos(5x))/(e^x)

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Apply Integration By Parts

: \frac{1}{5} e^{- x} sin (5 x) - \int - \frac{1}{5} e^{- x} sin (5 x) d x

= \frac{1}{5} e^{- x} sin (5 x) - \int - \frac{1}{5} e^{- x} sin (5 x) d x

Take the constant out:

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

= \frac{1}{5} e^{- x} sin (5 x) - (- \frac{1}{5} \cdot \int e^{- x} sin (5 x) d x)

Apply Integration By Parts

: - \frac{1}{5} e^{- x} cos (5 x) - \int \frac{1}{5} e^{- x} cos (5 x) d x

= \frac{1}{5} e^{- x} sin (5 x) - (- \frac{1}{5} (- \frac{1}{5} e^{- x} cos (5 x) - \int \frac{1}{5} e^{- x} cos (5 x) d x))

Take the constant out:

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

= \frac{1}{5} e^{- x} sin (5 x) - (- \frac{1}{5} (- \frac{1}{5} e^{- x} cos (5 x) - \frac{1}{5} \cdot \int e^{- x} cos (5 x) d x))

Therefore

\int e^{- x} cos (5 x) d x = \frac{1}{5} e^{- x} sin (5 x) - (- \frac{1}{5} (- \frac{1}{5} e^{- x} cos (5 x) - \frac{1}{5} \cdot \int e^{- x} cos (5 x) d x))

Isolate

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

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

Add a constant to the solution

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

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