5.4 Indefinite Integrals and the Net Change Theorem/1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int\frac{x}{\sqrt{x^2+1}}dx=\sqrt{x^2+1}+c}

Show that: ${\frac {d}{dx}}\left[(x^{2}+1)^{\frac {1}{2}}+c\right]={\frac {x}{\sqrt {x^{2}+1}}}$

${\begin{aligned}a&=x^{2}+1b&=a^{1/2}\\[0.6ex]{\frac {da}{dx}}&=2x&{\frac {db}{da}}&={\frac {1}{2}}a^{-1/2}\end{aligned}}$

${\frac {da}{dx}}\cdot {\frac {db}{da}}=\left(2x\right)\left({\frac {1}{2}}a^{-1/2}\right)=xa^{-1/2}=x(x^{2}+1)^{-1/2}={\frac {x}{\sqrt {x^{2}+1}}}$