What is the integral of e^{2x}e^{e^x} ?

The integral of e^{2x}e^{e^x} is e^{e^x+x}-e^{e^x}+C

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integral of e^{2x}e^{e^x}

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

e^{e^{x} + x} - e^{e^{x}} + C

Solution steps

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

Apply u-substitution

: \int e^{u} u d u

= \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= e^{u} u - \int e^{u} d u

\int e^{u} d u = e^{u}

= e^{u} u - e^{u}

Substitute back

u = e^{x}

= e^{e^{x}} e^{x} - e^{e^{x}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{e^{x}} e^{x} = e^{e^{x} + x}

= e^{e^{x} + x} - e^{e^{x}}

Add a constant to the solution

= e^{e^{x} + x} - e^{e^{x}} + C

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What is the integral of e^{2x}e^{e^x} ?
The integral of e^{2x}e^{e^x} is e^{e^x+x}-e^{e^x}+C