How do you find the integral of ${(1 + e^{2} x)}^{\frac{1}{2}}$ $(1 + e^2x) ^(1/2)$ ?

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How do you find the integral of ${(1 + e^{2} x)}^{\frac{1}{2}}$ $(1 + e^2x) ^(1/2)$ ?

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Answer 1 · 2015-10-06T17:44:41

$\int ({(1 + e^{2} x)}^{\frac{1}{2}}) d x = \frac{2}{3 e^{2}} {(1 + e^{2} x)}^{\frac{3}{2}}$ $int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(3/2)$

Explanation:

Integrate by method of substitution.

Solution:
(1) Let u = $\sqrt{1 + e^{2} x}$ $sqrt(1+e^2x$
(2) Take the square of u, hence, $u^{2} = 1 + e^{2} x$ $u^2=1+e^2x$
(3) Take the derivative of both sides, hence, $2 u d u = e^{2} d x$ $2udu=e^2dx$
(4) Substitute 'u' and 'du' to the original differential eqn.
$\int u \cdot 2 \frac{u}{e^{2}} d u$ $int u * 2u/e^2 du$
(5) Integrate, $\frac{2}{e^{2}} \int u^{2} d u = \frac{2}{3 e^{2}} u^{3}$ $2/e^2intu^2du=2/(3e^2)u^3$
(6) Replace 'u' in terms of 'x' by using the defined value of 'u'
$\int ({(1 + e^{2} x)}^{\frac{1}{2}}) d x = \frac{2}{3 e^{2}} {(1 + e^{2} x)}^{\frac{3}{2}}$ $int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(3/2)$

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How do you find the integral of ${(1 + e^{2} x)}^{\frac{1}{2}}$ $(1 + e^2x) ^(1/2)$ ?

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Explanation:

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