What is the integral of e^{5x}*sqrt(3e^{5x)+2} ?

The integral of e^{5x}*sqrt(3e^{5x)+2} is 2/45 (3e^{5x}+2)^{3/2}+C

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integral of e^{5x}*sqrt(3e^{5x)+2}

\int e^{5 x} \cdot 3 e^{5 x} + 2 d x

\frac{2}{45} (3 e^{5 x} + 2)^{\frac{3}{2}} + C

Solution steps

\int e^{5 x} 3 e^{5 x} + 2 d x

Apply u-substitution

: \int e^{u} 3 e^{u} + 2 \frac{1}{5} d u

= \int e^{u} 3 e^{u} + 2 \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int e^{u} 3 e^{u} + 2 d u

Apply u-substitution

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

= \frac{1}{5} \cdot \int \frac{v}{3} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{1}{3} \cdot \int v d v

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= \frac{1}{5} \cdot \frac{1}{3} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

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

Simplify

\frac{1}{5} \cdot \frac{1}{3} \cdot \frac{2}{3} (3 e^{5 x} + 2)^{\frac{3}{2}} : \frac{2}{45} (3 e^{5 x} + 2)^{\frac{3}{2}}

= \frac{2}{45} (3 e^{5 x} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{45} (3 e^{5 x} + 2)^{\frac{3}{2}} + C

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What is the integral of e^{5x}*sqrt(3e^{5x)+2} ?
The integral of e^{5x}*sqrt(3e^{5x)+2} is 2/45 (3e^{5x}+2)^{3/2}+C