What is the integral of e^xg(e^x) ?

The integral of e^xg(e^x) is (ge^{2x})/2+C

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integral of e^xg(e^x)

\int e^{x} g (e^{x}) d x

\frac{g e ^{2 x}}{2} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

e^{x} e^{x} = (ee)^{x}

= g \cdot \int (ee)^{x} d x

\int a^{x} d x = \frac{a ^{x}}{ln a}

= g \frac{( ee ) ^{x}}{ln ( ee )}

Simplify

g \frac{( ee ) ^{x}}{ln ( ee )} : \frac{g e ^{2 x}}{2}

= \frac{g e ^{2 x}}{2}

Add a constant to the solution

= \frac{g e ^{2 x}}{2} + C