Using the substitution $u = \sqrt{x}$ $u=sqrtx$ , how do you integrate $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}$ $int e^sqrtx/sqrtx$ from [1,4]?

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Using the substitution $u = \sqrt{x}$ $u=sqrtx$ , how do you integrate $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}$ $int e^sqrtx/sqrtx$ from [1,4]?

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Answer 1 · 2017-01-11T11:18:44

$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x = 2 e (e - 1)$ $int_1^4 e^sqrt(x)/sqrt(x) dx = 2e(e-1)$

Explanation:

If we substitute:

$u = \sqrt{x}$ $u = sqrt(x)$

we have:

$d u = \frac{d x}{2 \sqrt{x}}$ $du = (dx) / (2sqrt(x))$

and of course:

$x = 1 \Rightarrow u = 1$ $x=1 => u = 1$
$x = 4 \Rightarrow u = 2$ $x=4 => u=2$

So:

$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x = 2 \int_{1}^{2} e^{u} d u = {[e^{u}]}_{1}^{u} = 2 (e^{2} - e) = 2 e (e - 1)$ $int_1^4 e^sqrt(x)/sqrt(x) dx = 2 int_1^2 e^udu = [e^u]_1^2 = 2(e^2-e)= 2e(e-1)$

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Using the substitution $u = \sqrt{x}$ $u=sqrtx$ , how do you integrate $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}$ $int e^sqrtx/sqrtx$ from [1,4]?

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Using the substitution $u = \sqrt{x}$ $u=sqrtx$ , how do you integrate $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}$ $int e^sqrtx/sqrtx$ from [1,4]?