What is the integral of (2/(e^{3x)}+sin(4x)) ?

The integral of (2/(e^{3x)}+sin(4x)) is -2/3 e^{-3x}-1/4 cos(4x)+C

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integral of (2/(e^{3x)}+sin(4x))

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

- \frac{2}{3} e^{- 3 x} - \frac{1}{4} cos (4 x) + C

Solution steps

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

\int \frac{2}{e ^{3 x}} d x = - \frac{2}{3} e^{- 3 x}

\int sin (4 x) d x = - \frac{1}{4} cos (4 x)

= - \frac{2}{3} e^{- 3 x} - \frac{1}{4} cos (4 x)

Add a constant to the solution

= - \frac{2}{3} e^{- 3 x} - \frac{1}{4} cos (4 x) + C

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What is the integral of (2/(e^{3x)}+sin(4x)) ?
The integral of (2/(e^{3x)}+sin(4x)) is -2/3 e^{-3x}-1/4 cos(4x)+C