What is the integral of e^{3x}sqrt(1+e^{3x)} ?

The integral of e^{3x}sqrt(1+e^{3x)} is 2/9 (1+e^{3x})^{3/2}+C

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integral of e^{3x}sqrt(1+e^{3x)}

\int e^{3 x} 1 + e^{3 x} d x

\frac{2}{9} (1 + e^{3 x})^{\frac{3}{2}} + C

Solution steps

\int e^{3 x} 1 + e^{3 x} d x

Apply u-substitution

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

= \int \frac{u}{3} 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 u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + e^{3 x}

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

Simplify

\frac{1}{3} \cdot \frac{2}{3} (1 + e^{3 x})^{\frac{3}{2}} : \frac{2}{9} (1 + e^{3 x})^{\frac{3}{2}}

= \frac{2}{9} (1 + e^{3 x})^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (1 + e^{3 x})^{\frac{3}{2}} + C

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What is the integral of e^{3x}sqrt(1+e^{3x)} ?
The integral of e^{3x}sqrt(1+e^{3x)} is 2/9 (1+e^{3x})^{3/2}+C