積分記号 (r^2)/((1+r^2))

解

積分記号

\frac{r ^{2}}{( 1 + r ^{2} )}

- arctan (r) + r + C

解答ステップ

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

\frac{r ^{2}}{1 + r ^{2}} = - \frac{1}{1 + r ^{2}} + 1

= \int - \frac{1}{1 + r ^{2}} + 1 d r

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 + r ^{2}} d r + \int 1 d r

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

\int 1 d r = r

= - arctan (r) + r

定数を解答に追加する

= - arctan (r) + r + C