What is the integral of e^{2t+e^t} ?

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

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integral of e^{2t+e^t}

\int e^{2 t + e^{t}} d t

e^{e^{t} + t} - e^{e^{t}} + C

Solution steps

\int e^{2 t + e^{t}} d t

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^{t}

= e^{e^{t}} e^{t} - e^{e^{t}}

Apply exponent rule:

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

e^{e^{t}} e^{t} = e^{e^{t} + t}

= e^{e^{t} + t} - e^{e^{t}}

Add a constant to the solution

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

\frac{d}{d x} ((5 x^{2} - 2 x + 1)^{- 3})

x \to 3 lim (\frac{2 + x}{x - 3})

\frac{d}{d t} (2 t^{2} - 10)

\int - \frac{5}{3 x} d x

\frac{( 2 x ^{4} + 5 x ^{3} - 2 x - 8 )}{( x + 3 )}

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