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

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

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

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

- \frac{3}{2} e^{- 2 x} + \frac{1}{2} e^{2 x} + C

Solution steps

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

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} (e^{4 x} + 3) - \int - 2 e^{2 x} d x

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

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

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

Simplify

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

= - \frac{3}{2} e^{- 2 x} + \frac{1}{2} e^{2 x}

Add a constant to the solution

= - \frac{3}{2} e^{- 2 x} + \frac{1}{2} e^{2 x} + C

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