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

The integral of 3^{-x}cos(x) is (3^{-x}sin(x))/(1+ln^2(3))-(ln(3)*3^{-x}cos(x))/(1+ln^2(3))+C

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

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

\frac{3 ^{- x} sin ( x )}{1 + ln ^{2} ( 3 )} - \frac{ln ( 3 ) \cdot 3 ^{- x} cos ( x )}{1 + ln ^{2} ( 3 )} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{3 ^{- x} sin ( x )}{1 + ln ^{2} ( 3 )} - \frac{ln ( 3 ) \cdot 3 ^{- x} cos ( x )}{1 + ln ^{2} ( 3 )}

Add a constant to the solution

= \frac{3 ^{- x} sin ( x )}{1 + ln ^{2} ( 3 )} - \frac{ln ( 3 ) \cdot 3 ^{- x} cos ( x )}{1 + ln ^{2} ( 3 )} + C

\int 1 + \frac{1}{x} d x

\frac{\partial}{\partial y} (5 x^{2} + 8 x y + y^{2})

\frac{d y}{d x} + \frac{5 x ^{2}}{e ^{3 x^{3}}} y = 0

x y^{^{'}} - 2 y = 2 x^{2} + x - 2

\int \frac{( x ^{2} - x + 4 )}{x ^{3} + 2 x} d x

What is the integral of 3^{-x}cos(x) ?
The integral of 3^{-x}cos(x) is (3^{-x}sin(x))/(1+ln^2(3))-(ln(3)*3^{-x}cos(x))/(1+ln^2(3))+C