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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int u (u - 1) d u

= \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 = 1 + e^{x}

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

Add a constant to the solution

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

x \to 0 lim (\frac{x - sin ( x )}{x})

\frac{\partial}{\partial y} (y sin (y))

y = 8 x sin (x^{2})

\frac{\partial}{\partial y} (7 e^{7 e^{y x} + y} x)

\frac{d}{d x} (- 4 sin (2 x) + 3 cos (3 x))

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