What is the integral of e^(2x) + e^(-2x) with respect to x?

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To find the integral of ( e^{2x} + e^{-2x} ) with respect to ( x ), we can integrate each term separately.

The integral of ( e^{2x} ) is computed using the formula ( \int e^{ax} , dx = \frac{1}{a} e^{ax} + C ). Here, ( a = 2 ), so:

[

\int e^{2x} , dx = \frac{1}{2} e^{2x} + C_1

]

Similarly, for ( e^{-2x} ), we again use the integration formula where ( a = -2 ):

[

\int e^{-2x} , dx = \frac{1}{-2} e^{-2x} + C_2 = -\frac{1}{2} e^{-2x} + C_2

]

Putting these results together, the integral of ( e^{2x} + e^{-2x} ) becomes:

[

\int (e^{2x} + e^{-2x}) , dx = \left( \frac{1}{2} e