What is the integral of (20e^{5t})/(sqrt(5+e^{5t))} ?

The integral of (20e^{5t})/(sqrt(5+e^{5t))} is 8sqrt(5+e^{5t)}+C

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integral of (20e^{5t})/(sqrt(5+e^{5t))}

\int \frac{20 e ^{5 t}}{5 + e ^{5 t}} d t

8 5 + e^{5 t} + C

Solution steps

\int \frac{20 e ^{5 t}}{5 + e ^{5 t}} d t

Take the constant out:

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

= 20 \cdot \int \frac{e ^{5 t}}{5 + e ^{5 t}} d t

Apply u-substitution

: \int \frac{e ^{u}}{5 5 + e ^{u}} d u

= 20 \cdot \int \frac{e ^{u}}{5 5 + e ^{u}} d u

Take the constant out:

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

= 20 \cdot \frac{1}{5} \cdot \int \frac{e ^{u}}{5 + e ^{u}} d u

Apply u-substitution

: \int \frac{1}{v} d v

= 20 \cdot \frac{1}{5} \cdot \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= 20 \cdot \frac{1}{5} \cdot 2 v^{\frac{1}{2}}

Substitute back

= 20 \cdot \frac{1}{5} \cdot 2 (5 + e^{5 t})^{\frac{1}{2}}

Simplify

20 \cdot \frac{1}{5} \cdot 2 (5 + e^{5 t})^{\frac{1}{2}} : 8 5 + e^{5 t}

= 8 5 + e^{5 t}

Add a constant to the solution

= 8 5 + e^{5 t} + C

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What is the integral of (20e^{5t})/(sqrt(5+e^{5t))} ?
The integral of (20e^{5t})/(sqrt(5+e^{5t))} is 8sqrt(5+e^{5t)}+C