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

The integral of e^{2x}sqrt(e^x+2) is 2/5 (e^x+2)^{5/2}-4/3 (e^x+2)^{3/2}+C

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

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

\frac{2}{5} (e^{x} + 2)^{\frac{5}{2}} - \frac{4}{3} (e^{x} + 2)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

: \int u u + 2 d u

= \int u u + 2 d u

Apply u-substitution

: \int (v - 2) v d v

= \int (v - 2) v d v

Expand

(v - 2) v : v^{\frac{3}{2}} - 2 v

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int v^{\frac{3}{2}} d v = \frac{2}{5} v^{\frac{5}{2}}

\int 2 v d v = \frac{4}{3} v^{\frac{3}{2}}

= \frac{2}{5} v^{\frac{5}{2}} - \frac{4}{3} v^{\frac{3}{2}}

Substitute back

= \frac{2}{5} (e^{x} + 2)^{\frac{5}{2}} - \frac{4}{3} (e^{x} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (e^{x} + 2)^{\frac{5}{2}} - \frac{4}{3} (e^{x} + 2)^{\frac{3}{2}} + C

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What is the integral of e^{2x}sqrt(e^x+2) ?
The integral of e^{2x}sqrt(e^x+2) is 2/5 (e^x+2)^{5/2}-4/3 (e^x+2)^{3/2}+C