What is the integral of e^tcos(e^t) ?

The integral of e^tcos(e^t) is sin(e^t)+C

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integral of e^tcos(e^t)

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

sin (e^{t}) + C

Solution steps

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

Apply u-substitution

: \int cos (u) d u

= \int cos (u) d u

Use the common integral:

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

= sin (u)

Substitute back

u = e^{t}

= sin (e^{t})

Add a constant to the solution

= sin (e^{t}) + C

f (x) = - 5

y^{^{'}} = (x - 5) e^{- 2 y}, y (5) = ln (5)

y^{^{''}} - 12 y^{^{'}} + 36 y = 2 t e^{6 t}

t ln (t) \frac{d r}{d t} + r = 9 t e^{t}

\frac{d}{d x} ((\frac{x - 9}{x + 2})^{\frac{3}{2}})