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

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

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integral of e^{sqrt(5-3x)}

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{3} e^{u} d u

= \int - \frac{1}{3} e^{u} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int e^{u} d u

Apply u-substitution

: \int e^{v} \cdot 2 v d v

= - \frac{1}{3} \cdot \int e^{v} \cdot 2 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 2 \cdot \int e^{v} v d v

Apply Integration By Parts

: e^{v} v - \int e^{v} d v

= - \frac{1}{3} \cdot 2 (e^{v} v - \int e^{v} d v)

\int e^{v} d v = e^{v}

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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