Question

lf $y=\frac{\log (x+\sqrt{1+x^2})}{\sqrt{1+x^2}}$ then $(1+x^2)y_1+xy=$

• A. $y$
• B. $-y$
• C. $0$
• D. $1$

Answer 1

The correct option is D $1$

$y\sqrt{1+x^2}=\log (x+\sqrt{1+x^2})$
Differentiate both sides:
$y_1\sqrt{1+x^2}+\frac{yx}{\sqrt{1+x^2}}=\frac{1+\frac{x}{\sqrt{1+x^2}}}{x+\sqrt{1+x^2}}$
$\Rightarrow y_1\sqrt{1+x^2}+\frac{yx}{\sqrt{1+x^2}}=\frac{1}{\sqrt{1+x^2}}$

$\Rightarrow y_1(1+x^2)+yx=1$