What is the integral of 1/2 e^{-4x}+sin(x) ?

The integral of 1/2 e^{-4x}+sin(x) is -1/8 e^{-4x}-cos(x)+C

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integral of 1/2 e^{-4x}+sin(x)

\int \frac{1}{2} e^{- 4 x} + sin (x) d x

- \frac{1}{8} e^{- 4 x} - cos (x) + C

Solution steps

\int \frac{1}{2} e^{- 4 x} + sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{2} e^{- 4 x} d x + \int sin (x) d x

\int \frac{1}{2} e^{- 4 x} d x = - \frac{1}{8} e^{- 4 x}

\int sin (x) d x = - cos (x)

= - \frac{1}{8} e^{- 4 x} - cos (x)

Add a constant to the solution

= - \frac{1}{8} e^{- 4 x} - cos (x) + C

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What is the integral of 1/2 e^{-4x}+sin(x) ?
The integral of 1/2 e^{-4x}+sin(x) is -1/8 e^{-4x}-cos(x)+C