What is the integral of 2^x(2^{-x}+1) ?

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

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integral of 2^x(2^{-x}+1)

\int 2^{x} (2^{- x} + 1) d x

x + \frac{2 ^{x}}{ln ( 2 )} + C

Solution steps

\int 2^{x} (2^{- x} + 1) d x

Expand

2^{x} (2^{- x} + 1) : 1 + 2^{x}

= \int 1 + 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 1 d x + \int 2^{x} d x

\int 1 d x = x

\int 2^{x} d x = \frac{2 ^{x}}{ln ( 2 )}

= x + \frac{2 ^{x}}{ln ( 2 )}

Add a constant to the solution

= x + \frac{2 ^{x}}{ln ( 2 )} + C

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