log-2-x-2-dx-

Question Number 27282 by hp killer last updated on 04/Jan/18

\int l o g (2 + x^{2}) d x

Commented by abdo imad last updated on 04/Jan/18

i f y o u m e a n l n i n t e g r a t e p a r p a r t s u^{^{'}} = 1 a n d v = l n (2 + x^{2})

\int l n (2 + x^{2}) d x = x l n (2 + x^{2}) - \int x \frac{2 x}{2 + x^{2}} d x

= x l n (2 + x^{2}) - 2 \int \frac{x^{2}}{2 + x^{2}} d x

= x l n (2 + x^{2}) - 2 \int \frac{2 + x^{2} - 2}{2 + x^{2}} d x

= x l n (2 + x^{2}) - 2 x + 4 \int \frac{d x}{2 + x^{2}} a n d b y t h e c h a n g e m e n t x = \sqrt{2} t

\int \frac{d x}{2 + x^{2}} = \int \frac{\sqrt{2} d t}{2 + 2 t^{2}} = \frac{\sqrt{2}}{2} a r t a n (\frac{x}{\sqrt{2}}) s o

\int l n (2 + x^{2}) d x = x l n (2 + x^{2}) - 2 x + 2 \sqrt{2} a r c t a n (\frac{x}{\sqrt{2}}) .