What is the integral of (15e^{3t})/(sqrt(10+e^{3t))} ?

The integral of (15e^{3t})/(sqrt(10+e^{3t))} is 10sqrt(10+e^{3t)}+C

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integral of (15e^{3t})/(sqrt(10+e^{3t))}

\int \frac{15 e ^{3 t}}{10 + e ^{3 t}} d t

10 10 + e^{3 t} + C

Solution steps

\int \frac{15 e ^{3 t}}{10 + e ^{3 t}} d t

Take the constant out:

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

= 15 \cdot \int \frac{e ^{3 t}}{10 + e ^{3 t}} d t

Apply u-substitution

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

= 15 \cdot \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

= 15 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 15 \cdot \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 10 + e^{3 t}

= 15 \cdot \frac{1}{3} \cdot 2 (10 + e^{3 t})^{\frac{1}{2}}

Simplify

15 \cdot \frac{1}{3} \cdot 2 (10 + e^{3 t})^{\frac{1}{2}} : 10 10 + e^{3 t}

= 10 10 + e^{3 t}

Add a constant to the solution

= 10 10 + e^{3 t} + C

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What is the integral of (15e^{3t})/(sqrt(10+e^{3t))} ?
The integral of (15e^{3t})/(sqrt(10+e^{3t))} is 10sqrt(10+e^{3t)}+C