How do you evaluate the integral $\int e^{2 x} \sqrt{1 + e^{2 x}}$ $int e^(2x)sqrt(1+e^(2x))$ ?

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How do you evaluate the integral $\int e^{2 x} \sqrt{1 + e^{2 x}}$ $int e^(2x)sqrt(1+e^(2x))$ ?

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Answer 1 · 2017-03-01T20:21:24

$\frac{1}{3} {(1 + e^{2 x})}^{\frac{3}{2}} + C$ $1/3(1 + e^(2x))^(3/2) + C$

Explanation:

Let $u = 1 + e^{2 x}$ $u = 1 + e^(2x)$ . Then $d u = 2 e^{2 x} d x$ $du = 2e^(2x)dx$ and $d x = \frac{d u}{2 e^{2 x}}$ $dx = (du)/(2e^(2x))$ .

$\int e^{2 x} \sqrt{u} \cdot \frac{d u}{2 e^{2 x}}$ $int e^(2x)sqrt(u) * (du)/(2e^(2x))$

$\frac{1}{2} \int \sqrt{u} d u$ $1/2int sqrt(u) du$

$\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}}) + C$ $1/2(2/3u^(3/2)) + C$

$\frac{1}{3} u^{\frac{3}{2}} + C$ $1/3u^(3/2) + C$

$\frac{1}{3} {(1 + e^{2 x})}^{\frac{3}{2}} + C$ $1/3(1 + e^(2x))^(3/2) + C$

Hopefully this helps!

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How do you evaluate the integral $\int e^{2 x} \sqrt{1 + e^{2 x}}$ $int e^(2x)sqrt(1+e^(2x))$ ?

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How do you evaluate the integral $\int e^{2 x} \sqrt{1 + e^{2 x}}$ $int e^(2x)sqrt(1+e^(2x))$ ?