What is the integral of (e^tcos(e^t))/(3+3sin(e^t)) ?

The integral of (e^tcos(e^t))/(3+3sin(e^t)) is 1/3 ln|3+3sin(e^t)|+C

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integral of (e^tcos(e^t))/(3+3sin(e^t))

\int \frac{e ^{t} cos ( e ^{t} )}{3 + 3 sin ( e ^{t} )} d t

\frac{1}{3} ln 3 + 3 sin (e^{t}) + C

Solution steps

\int \frac{e ^{t} cos ( e ^{t} )}{3 + 3 sin ( e ^{t} )} d t

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 3 + 3 sin (e^{t})

= \frac{1}{3} ln 3 + 3 sin (e^{t})

Add a constant to the solution

= \frac{1}{3} ln 3 + 3 sin (e^{t}) + C

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What is the integral of (e^tcos(e^t))/(3+3sin(e^t)) ?
The integral of (e^tcos(e^t))/(3+3sin(e^t)) is 1/3 ln|3+3sin(e^t)|+C