What is the integral of e^xsqrt(1+2e^x) ?

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

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integral of e^xsqrt(1+2e^x)

\int e^{x} 1 + 2 e^{x} d x

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

Solution steps

\int e^{x} 1 + 2 e^{x} d x

Apply u-substitution

= \int \frac{u}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = 1 + 2 e^{x}

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

Simplify

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

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

Add a constant to the solution

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

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