What is the integral of (e^{2t})/(e^t+3) ?

The integral of (e^{2t})/(e^t+3) is e^t+3-3ln|e^t+3|+C

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integral of (e^{2t})/(e^t+3)

\int \frac{e ^{2 t}}{e ^{t} + 3} d t

e^{t} + 3 - 3 ln e^{t} + 3 + C

Solution steps

\int \frac{e ^{2 t}}{e ^{t} + 3} d t

Apply u-substitution

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

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

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= u - 3 ln ∣ u ∣

Substitute back

u = e^{t} + 3

= e^{t} + 3 - 3 ln e^{t} + 3

Add a constant to the solution

= e^{t} + 3 - 3 ln e^{t} + 3 + C

\frac{d y}{d x} + 6 y = 4

2 cos (2 t)

e^{y - 1}

x \to 2 - lim (\frac{∣ x - 2 ∣}{x})

\frac{d y}{d x} = x + \frac{3 y}{x}

What is the integral of (e^{2t})/(e^t+3) ?
The integral of (e^{2t})/(e^t+3) is e^t+3-3ln|e^t+3|+C