integral of (x+3)^2(2-x)^5

解

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

- \frac{x ^{8}}{8} + \frac{4 x ^{7}}{7} + \frac{11 x ^{6}}{6} - 14 x^{5} + 10 x^{4} + \frac{272 x ^{3}}{3} - 264 x^{2} + 288 x + C

解答ステップ

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

拡張

(x + 3)^{2} (2 - x)^{5} : - x^{7} + 4 x^{6} + 11 x^{5} - 70 x^{4} + 40 x^{3} + 272 x^{2} - 528 x + 288

= \int - x^{7} + 4 x^{6} + 11 x^{5} - 70 x^{4} + 40 x^{3} + 272 x^{2} - 528 x + 288 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 x^{7} d x + \int 4 x^{6} d x + \int 11 x^{5} d x - \int 70 x^{4} d x + \int 40 x^{3} d x + \int 272 x^{2} d x - \int 528 x d x + \int 288 d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 4 x^{6} d x = \frac{4 x ^{7}}{7}

\int 11 x^{5} d x = \frac{11 x ^{6}}{6}

\int 70 x^{4} d x = 14 x^{5}

\int 40 x^{3} d x = 10 x^{4}

\int 272 x^{2} d x = \frac{272 x ^{3}}{3}

\int 528 x d x = 264 x^{2}

\int 288 d x = 288 x

= - \frac{x ^{8}}{8} + \frac{4 x ^{7}}{7} + \frac{11 x ^{6}}{6} - 14 x^{5} + 10 x^{4} + \frac{272 x ^{3}}{3} - 264 x^{2} + 288 x

定数を解答に追加する

= - \frac{x ^{8}}{8} + \frac{4 x ^{7}}{7} + \frac{11 x ^{6}}{6} - 14 x^{5} + 10 x^{4} + \frac{272 x ^{3}}{3} - 264 x^{2} + 288 x + C