What is the integral of (28e^{4t})/(sqrt(6+e^{4t))} ?

The integral of (28e^{4t})/(sqrt(6+e^{4t))} is 14sqrt(6+e^{4t)}+C

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integral of (28e^{4t})/(sqrt(6+e^{4t))}

\int \frac{28 e ^{4 t}}{6 + e ^{4 t}} d t

14 6 + e^{4 t} + C

Solution steps

\int \frac{28 e ^{4 t}}{6 + e ^{4 t}} d t

Take the constant out:

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

= 28 \cdot \int \frac{e ^{4 t}}{6 + e ^{4 t}} d t

Apply u-substitution

: \int \frac{1}{4 u} d u

= 28 \cdot \int \frac{1}{4 u} d u

Take the constant out:

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

= 28 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 28 \cdot \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 6 + e^{4 t}

= 28 \cdot \frac{1}{4} \cdot 2 (6 + e^{4 t})^{\frac{1}{2}}

Simplify

28 \cdot \frac{1}{4} \cdot 2 (6 + e^{4 t})^{\frac{1}{2}} : 14 6 + e^{4 t}

= 14 6 + e^{4 t}

Add a constant to the solution

= 14 6 + e^{4 t} + C

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What is the integral of (28e^{4t})/(sqrt(6+e^{4t))} ?
The integral of (28e^{4t})/(sqrt(6+e^{4t))} is 14sqrt(6+e^{4t)}+C