How do you use substitution to integrate $f (x) = \frac{x}{\sqrt{20 + x^{2}}}$ $f(x)= x/sqrt(20+x^2)$ ?

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How do you use substitution to integrate $f (x) = \frac{x}{\sqrt{20 + x^{2}}}$ $f(x)= x/sqrt(20+x^2)$ ?

1 Answer

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Answer 1 · 2018-03-12T09:30:02

$\int \frac{x}{\sqrt{20 + x^{2}}} d x = \sqrt{20 + x^{2}} + C$ $int x/sqrt(20+x^2)dx = sqrt(20+x^2)+C$

Explanation:

Substitute $u = 20 + x^{2}$ $u = 20+x^2$ , $d u = 2 x d x$ $du = 2xdx$

$\int \frac{x}{\sqrt{20 + x^{2}}} d x = \frac{1}{2} \int \frac{2 x d x}{\sqrt{20 + x^{2}}} = \int \frac{d u}{2 \sqrt{u}} = \sqrt{u} + C$ $int x/sqrt(20+x^2)dx = 1/2 int (2xdx)/sqrt(20+x^2) = int (du)/(2sqrtu) = sqrt(u)+C$

and undoing the substitution:

$\int \frac{x}{\sqrt{20 + x^{2}}} d x = \sqrt{20 + x^{2}} + C$ $int x/sqrt(20+x^2)dx = sqrt(20+x^2)+C$

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How do you use substitution to integrate $f (x) = \frac{x}{\sqrt{20 + x^{2}}}$ $f(x)= x/sqrt(20+x^2)$ ?

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Explanation:

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How do you use substitution to integrate $f (x) = \frac{x}{\sqrt{20 + x^{2}}}$ $f(x)= x/sqrt(20+x^2)$ ?