What is the integral of e^{2x}sqrt(1+4e^x) ?

The integral of e^{2x}sqrt(1+4e^x) is 1/60 (6e^x-1)(4e^x+1)^{3/2}+C

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integral of e^{2x}sqrt(1+4e^x)

\int e^{2 x} 1 + 4 e^{x} d x

\frac{1}{60} (6 e^{x} - 1) (4 e^{x} + 1)^{\frac{3}{2}} + C

Solution steps

\int e^{2 x} 1 + 4 e^{x} d x

Apply u-substitution

= \int \frac{u ( u - 1 )}{16} d u

Take the constant out:

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

= \frac{1}{16} \cdot \int u (u - 1) d u

Expand

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

= \frac{1}{16} \cdot \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

= \frac{1}{16} (\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{1}{16} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 + 4 e^{x}

= \frac{1}{16} (\frac{2}{5} (1 + 4 e^{x})^{\frac{5}{2}} - \frac{2}{3} (1 + 4 e^{x})^{\frac{3}{2}})

Simplify

\frac{1}{16} (\frac{2}{5} (1 + 4 e^{x})^{\frac{5}{2}} - \frac{2}{3} (1 + 4 e^{x})^{\frac{3}{2}}) : \frac{1}{60} (6 e^{x} - 1) (4 e^{x} + 1)^{\frac{3}{2}}

= \frac{1}{60} (6 e^{x} - 1) (4 e^{x} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{60} (6 e^{x} - 1) (4 e^{x} + 1)^{\frac{3}{2}} + C

What is the integral of e^{2x}sqrt(1+4e^x) ?
The integral of e^{2x}sqrt(1+4e^x) is 1/60 (6e^x-1)(4e^x+1)^{3/2}+C