What is the integral of (2e^{2t})/(5+e^{2t)} ?

The integral of (2e^{2t})/(5+e^{2t)} is ln|5+e^{2t}|+C

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integral of (2e^{2t})/(5+e^{2t)}

\int \frac{2 e ^{2 t}}{5 + e ^{2 t}} d t

ln 5 + e^{2 t} + C

Solution steps

\int \frac{2 e ^{2 t}}{5 + e ^{2 t}} d t

Take the constant out:

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

= 2 \cdot \int \frac{e ^{2 t}}{5 + e ^{2 t}} d t

Apply u-substitution

: \int \frac{1}{2 u} d u

= 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 5 + e^{2 t}

= 2 \cdot \frac{1}{2} ln 5 + e^{2 t}

Simplify

2 \cdot \frac{1}{2} ln 5 + e^{2 t} : ln 5 + e^{2 t}

= ln 5 + e^{2 t}

Add a constant to the solution

= ln 5 + e^{2 t} + C

64 x^{2}

ln (2 x) 4

\int_{0}^{\frac{π}{2}} cos^{2} (\frac{x}{2}) d x

(3 sin (t))^{^{'}}

x^{3} - 3 x^{2} + 5 x - 7

What is the integral of (2e^{2t})/(5+e^{2t)} ?
The integral of (2e^{2t})/(5+e^{2t)} is ln|5+e^{2t}|+C