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$∫ \dfrac{g(x)g'(x)}{\sqrt{(1+g^2(x))}}$ where $g'(x)$ is continous. I tried use use substitution but I couldn't figure out which one to substitute.

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    Try substituting $u=g(x)$ to make it more readable. Then try $v = 1+u^2$. – 2017-02-10

2 Answers 2

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I am attaching the solution of the pic. and go through it. Although you got enough hint.

enter image description here

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$g(x)=t$, then you get

$\int \frac{t}{\sqrt{1+t^{2}}}dt=\sqrt{1+t^{2}}+C$,

so

$\int \frac{g(x)g'(x)}{\sqrt{1+g(x)^{2}}}dx=\sqrt{1+g(x)^{2}}+C$