What is the integral of (9e^x-3e^{-x})^2 ?

The integral of (9e^x-3e^{-x})^2 is 81/2 e^{2x}-54x-9/2 e^{-2x}+C

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integral of (9e^x-3e^{-x})^2

\int (9 e^{x} - 3 e^{- x})^{2} d x

\frac{81}{2} e^{2 x} - 54 x - \frac{9}{2} e^{- 2 x} + C

Solution steps

\int (9 e^{x} - 3 e^{- x})^{2} d x

Expand

(9 e^{x} - 3 e^{- x})^{2} : 81 e^{2 x} - 54 + 9 e^{- 2 x}

= \int 81 e^{2 x} - 54 + 9 e^{- 2 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 81 e^{2 x} d x - \int 54 d x + \int 9 e^{- 2 x} d x

\int 81 e^{2 x} d x = \frac{81}{2} e^{2 x}

\int 54 d x = 54 x

\int 9 e^{- 2 x} d x = - \frac{9}{2} e^{- 2 x}

= \frac{81}{2} e^{2 x} - 54 x - \frac{9}{2} e^{- 2 x}

Add a constant to the solution

= \frac{81}{2} e^{2 x} - 54 x - \frac{9}{2} e^{- 2 x} + C

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What is the integral of (9e^x-3e^{-x})^2 ?
The integral of (9e^x-3e^{-x})^2 is 81/2 e^{2x}-54x-9/2 e^{-2x}+C