What is the integral of (e^{5x}-2e^{4x}-1)/(e^{4x)} ?

The integral of (e^{5x}-2e^{4x}-1)/(e^{4x)} is e^x-2x+1/4 e^{-4x}+C

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integral of (e^{5x}-2e^{4x}-1)/(e^{4x)}

\int \frac{e ^{5 x} - 2 e ^{4 x} - 1}{e ^{4 x}} d x

e^{x} - 2 x + \frac{1}{4} e^{- 4 x} + C

Solution steps

\int \frac{e ^{5 x} - 2 e ^{4 x} - 1}{e ^{4 x}} d x

Expand

\frac{e ^{5 x} - 2 e ^{4 x} - 1}{e ^{4 x}} : e^{x} - 2 - \frac{1}{e ^{4 x}}

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

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

\int 2 d x = 2 x

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

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

Simplify

= e^{x} - 2 x + \frac{1}{4} e^{- 4 x}

Add a constant to the solution

= e^{x} - 2 x + \frac{1}{4} e^{- 4 x} + C

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What is the integral of (e^{5x}-2e^{4x}-1)/(e^{4x)} ?
The integral of (e^{5x}-2e^{4x}-1)/(e^{4x)} is e^x-2x+1/4 e^{-4x}+C