What is the integral of e^{-x}-e^{-3x} ?

The integral of e^{-x}-e^{-3x} is -e^{-x}+1/3 e^{-3x}+C

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integral of e^{-x}-e^{-3x}

\int e^{- x} - e^{- 3 x} d x

- e^{- x} + \frac{1}{3} e^{- 3 x} + C

Solution steps

\int e^{- x} - e^{- 3 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 e^{- x} d x - \int e^{- 3 x} d x

\int e^{- x} d x = - e^{- x}

\int e^{- 3 x} d x = - \frac{1}{3} e^{- 3 x}

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

Simplify

= - e^{- x} + \frac{1}{3} e^{- 3 x}

Add a constant to the solution

= - e^{- x} + \frac{1}{3} e^{- 3 x} + C

f (x) = 5 e^{x} cos (x), (0, 5)

\int_{0}^{2} y^{\frac{7}{2}} d y

\frac{d}{d x} (\frac{5 x}{x})

f (t) = t e^{2 t}

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

What is the integral of e^{-x}-e^{-3x} ?
The integral of e^{-x}-e^{-3x} is -e^{-x}+1/3 e^{-3x}+C