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

The integral of e^{-x/2}cos(3x) is (12e^{-x/2}sin(3x))/(37)-(2e^{-x/2}cos(3x))/(37)+C

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

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

\frac{12 e ^{- \frac{x}{2}} sin ( 3 x )}{37} - \frac{2 e ^{- \frac{x}{2}} cos ( 3 x )}{37} + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - \int - \frac{1}{6} e^{- \frac{x}{2}} sin (3 x) d x

= \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - \int - \frac{1}{6} e^{- \frac{x}{2}} sin (3 x) d x

Take the constant out:

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

= \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - (- \frac{1}{6} \cdot \int e^{- \frac{x}{2}} sin (3 x) d x)

Apply Integration By Parts

: - \frac{1}{3} e^{- \frac{x}{2}} cos (3 x) - \int \frac{1}{6} e^{- \frac{x}{2}} cos (3 x) d x

= \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - (- \frac{1}{6} (- \frac{1}{3} e^{- \frac{x}{2}} cos (3 x) - \int \frac{1}{6} e^{- \frac{x}{2}} cos (3 x) d x))

Take the constant out:

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

= \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - (- \frac{1}{6} (- \frac{1}{3} e^{- \frac{x}{2}} cos (3 x) - \frac{1}{6} \cdot \int e^{- \frac{x}{2}} cos (3 x) d x))

Therefore

\int e^{- \frac{x}{2}} cos (3 x) d x = \frac{1}{3} e^{- \frac{x}{2}} sin (3 x) - (- \frac{1}{6} (- \frac{1}{3} e^{- \frac{x}{2}} cos (3 x) - \frac{1}{6} \cdot \int e^{- \frac{x}{2}} cos (3 x) d x))

Isolate

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

= \frac{12 e ^{- \frac{x}{2}} sin ( 3 x )}{37} - \frac{2 e ^{- \frac{x}{2}} cos ( 3 x )}{37}

Add a constant to the solution

= \frac{12 e ^{- \frac{x}{2}} sin ( 3 x )}{37} - \frac{2 e ^{- \frac{x}{2}} cos ( 3 x )}{37} + C

\int \frac{x ^{3}}{x ^{2} - 9} d x

\frac{\partial}{\partial y} (6 + x ln (x y - 7))

\frac{\partial}{\partial x} (x y (1 - x - y))

\frac{d}{d x} (\frac{2}{sin ( x )})

What is the integral of e^{-x/2}cos(3x) ?
The integral of e^{-x/2}cos(3x) is (12e^{-x/2}sin(3x))/(37)-(2e^{-x/2}cos(3x))/(37)+C