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

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

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integral of sqrt(2+e^x)e^x

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

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

Solution steps

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

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

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

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

Substitute back

u = 2 + e^{x}

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

Add a constant to the solution

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

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