integral of ((2+x^2))/(1+x^2)

解

\int \frac{( 2 + x ^{2} )}{1 + x ^{2}} d x

arctan (x) + x + C

解答ステップ

\int \frac{2 + x ^{2}}{1 + x ^{2}} d x

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

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{1 + x ^{2}} d x = arctan (x)

\int 1 d x = x

= arctan (x) + x

定数を解答に追加する

= arctan (x) + x + C