integral of (r^3+5)^2

解

\int (r^{3} + 5)^{2} d r

\frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r + C

解答ステップ

\int (r^{3} + 5)^{2} d r

拡張

(r^{3} + 5)^{2} : r^{6} + 10 r^{3} + 25

= \int r^{6} + 10 r^{3} + 25 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 r^{6} d r + \int 10 r^{3} d r + \int 25 d r

\int r^{6} d r = \frac{r ^{7}}{7}

\int 10 r^{3} d r = \frac{5 r ^{4}}{2}

\int 25 d r = 25 r

= \frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r

定数を解答に追加する

= \frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r + C