prove-that-dx-1-x-2-ln-x-1-x-2-c-2-find-dx-a-x-2-with-a-gt-0-

Question Number 38124 by maxmathsup by imad last updated on 22/Jun/18

p r o v e t h a t \int \frac{d x}{\sqrt{1 + x^{2}}} = l n (x + \sqrt{1 + x^{2}}) + c

2) f i n d \int \frac{d x}{\sqrt{a + x^{2}}} w i t h a > 0

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18

y = l n (x + \sqrt{1 + x^{2})} + c

\frac{d y}{d x} = \frac{1}{x + \sqrt{1 + x^{2}}} \times (1 + \frac{2 x}{2 \sqrt{1 + x^{2}}})

= \frac{1}{x + \sqrt{1 + x^{2}}} \times \frac{x + \sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}

= \frac{1}{\sqrt{1 + x^{2}}}

d y = \frac{d x}{\sqrt{1 + x^{2}}}

y = \int \frac{d x}{\sqrt{1 + x^{2}}}