What is the integral of e^{2x}sqrt(e^x+1) ?

The integral of e^{2x}sqrt(e^x+1) is 2/5 (e^x+1)^{5/2}-2/3 (e^x+1)^{3/2}+C

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integral of e^{2x}sqrt(e^x+1)

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

\frac{2}{5} (e^{x} + 1)^{\frac{5}{2}} - \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

= \int u (u - 1) d u

Expand

u (u - 1) : u^{\frac{3}{2}} - u

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

Apply the Sum Rule:

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

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

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

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

= \frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = e^{x} + 1

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

Add a constant to the solution

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

What is the integral of e^{2x}sqrt(e^x+1) ?
The integral of e^{2x}sqrt(e^x+1) is 2/5 (e^x+1)^{5/2}-2/3 (e^x+1)^{3/2}+C