integral of (x^{20}-5x^4)^5(x^{19}-x^3)

解

\int (x^{20} - 5 x^{4})^{5} (x^{19} - x^{3}) d x

\frac{x ^{120}}{120} - \frac{x ^{104}}{4} + \frac{25 x ^{88}}{8} - \frac{125 x ^{72}}{6} + \frac{625 x ^{56}}{8} - \frac{625 x ^{40}}{4} + \frac{3125 x ^{24}}{24} + C

解答ステップ

\int (x^{20} - 5 x^{4})^{5} (x^{19} - x^{3}) d x

拡張

(x^{20} - 5 x^{4})^{5} (x^{19} - x^{3}) : x^{119} - 26 x^{103} + 275 x^{87} - 1500 x^{71} + 4375 x^{55} - 6250 x^{39} + 3125 x^{23}

= \int x^{119} - 26 x^{103} + 275 x^{87} - 1500 x^{71} + 4375 x^{55} - 6250 x^{39} + 3125 x^{23} 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^{119} d x - \int 26 x^{103} d x + \int 275 x^{87} d x - \int 1500 x^{71} d x + \int 4375 x^{55} d x - \int 6250 x^{39} d x + \int 3125 x^{23} d x

\int x^{119} d x = \frac{x ^{120}}{120}

\int 26 x^{103} d x = \frac{x ^{104}}{4}

\int 275 x^{87} d x = \frac{25 x ^{88}}{8}

\int 1500 x^{71} d x = \frac{125 x ^{72}}{6}

\int 4375 x^{55} d x = \frac{625 x ^{56}}{8}

\int 6250 x^{39} d x = \frac{625 x ^{40}}{4}

\int 3125 x^{23} d x = \frac{3125 x ^{24}}{24}

= \frac{x ^{120}}{120} - \frac{x ^{104}}{4} + \frac{25 x ^{88}}{8} - \frac{125 x ^{72}}{6} + \frac{625 x ^{56}}{8} - \frac{625 x ^{40}}{4} + \frac{3125 x ^{24}}{24}

定数を解答に追加する

= \frac{x ^{120}}{120} - \frac{x ^{104}}{4} + \frac{25 x ^{88}}{8} - \frac{125 x ^{72}}{6} + \frac{625 x ^{56}}{8} - \frac{625 x ^{40}}{4} + \frac{3125 x ^{24}}{24} + C