What is the integral of e^{2t}cos(e^t) ?

The integral of e^{2t}cos(e^t) is e^tsin(e^t)+cos(e^t)+C

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integral of e^{2t}cos(e^t)

\int e^{2 t} cos (e^{t}) d t

e^{t} sin (e^{t}) + cos (e^{t}) + C

Solution steps

\int e^{2 t} cos (e^{t}) d t

Apply u-substitution

: \int u cos (u) d u

= \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= u sin (u) - \int sin (u) d u

\int sin (u) d u = - cos (u)

= u sin (u) - (- cos (u))

Substitute back

u = e^{t}

= e^{t} sin (e^{t}) - (- cos (e^{t}))

Simplify

= e^{t} sin (e^{t}) + cos (e^{t})

Add a constant to the solution

= e^{t} sin (e^{t}) + cos (e^{t}) + C

x \to + 3 + lim (\frac{x + 9}{x - 3})

\frac{d}{d x} (3 tan^{2} (x))

\frac{r}{r ^{2} + 9}

x \to 0 lim (\frac{8}{x ^{2} + x} - \frac{8}{x})

What is the integral of e^{2t}cos(e^t) ?
The integral of e^{2t}cos(e^t) is e^tsin(e^t)+cos(e^t)+C