What is the integral of (e^x)/(sqrt(3e^x+2)) ?

The integral of (e^x)/(sqrt(3e^x+2)) is 2/3 sqrt(3e^x+2)+C

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integral of (e^x)/(sqrt(3e^x+2))

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

\frac{2}{3} 3 e^{x} + 2 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u + 2} d u

= \int \frac{1}{3 u + 2} d u

Apply u-substitution

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

= \int \frac{1}{3 v} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v} d v

Apply the Power Rule

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

= \frac{1}{3} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{3} \cdot 2 (3 e^{x} + 2)^{\frac{1}{2}}

Simplify

\frac{1}{3} \cdot 2 (3 e^{x} + 2)^{\frac{1}{2}} : \frac{2}{3} 3 e^{x} + 2

= \frac{2}{3} 3 e^{x} + 2

Add a constant to the solution

= \frac{2}{3} 3 e^{x} + 2 + C

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What is the integral of (e^x)/(sqrt(3e^x+2)) ?
The integral of (e^x)/(sqrt(3e^x+2)) is 2/3 sqrt(3e^x+2)+C