What is the integral of (e^x)/(sqrt(3-5e^x)) ?

The integral of (e^x)/(sqrt(3-5e^x)) is -2/5 sqrt(3-5e^x)+C

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integral of (e^x)/(sqrt(3-5e^x))

\int \frac{e ^{x}}{3 - 5 e ^{x}} d x

- \frac{2}{5} 3 - 5 e^{x} + C

Solution steps

\int \frac{e ^{x}}{3 - 5 e ^{x}} d x

Apply u-substitution

: \int \frac{1}{3 - 5 u} d u

= \int \frac{1}{3 - 5 u} d u

Apply u-substitution

: \int - \frac{1}{5 v} d v

= \int - \frac{1}{5 v} d v

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

= - \frac{1}{5} \cdot 2 (3 - 5 e^{x})^{\frac{1}{2}}

Simplify

- \frac{1}{5} \cdot 2 (3 - 5 e^{x})^{\frac{1}{2}} : - \frac{2}{5} 3 - 5 e^{x}

= - \frac{2}{5} 3 - 5 e^{x}

Add a constant to the solution

= - \frac{2}{5} 3 - 5 e^{x} + C

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What is the integral of (e^x)/(sqrt(3-5e^x)) ?
The integral of (e^x)/(sqrt(3-5e^x)) is -2/5 sqrt(3-5e^x)+C