integral of x^3(2-5x^4)^5

解

\int x^{3} (2 - 5 x^{4})^{5} d x

8 x^{4} - 50 x^{8} + \frac{500 x ^{12}}{3} - \frac{625 x ^{16}}{2} + \frac{625 x ^{20}}{2} - \frac{3125 x ^{24}}{24} + C

解答ステップ

\int x^{3} (2 - 5 x^{4})^{5} d x

拡張

x^{3} (2 - 5 x^{4})^{5} : 32 x^{3} - 400 x^{7} + 2000 x^{11} - 5000 x^{15} + 6250 x^{19} - 3125 x^{23}

= \int 32 x^{3} - 400 x^{7} + 2000 x^{11} - 5000 x^{15} + 6250 x^{19} - 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 32 x^{3} d x - \int 400 x^{7} d x + \int 2000 x^{11} d x - \int 5000 x^{15} d x + \int 6250 x^{19} d x - \int 3125 x^{23} d x

\int 32 x^{3} d x = 8 x^{4}

\int 400 x^{7} d x = 50 x^{8}

\int 2000 x^{11} d x = \frac{500 x ^{12}}{3}

\int 5000 x^{15} d x = \frac{625 x ^{16}}{2}

\int 6250 x^{19} d x = \frac{625 x ^{20}}{2}

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

= 8 x^{4} - 50 x^{8} + \frac{500 x ^{12}}{3} - \frac{625 x ^{16}}{2} + \frac{625 x ^{20}}{2} - \frac{3125 x ^{24}}{24}

定数を解答に追加する

= 8 x^{4} - 50 x^{8} + \frac{500 x ^{12}}{3} - \frac{625 x ^{16}}{2} + \frac{625 x ^{20}}{2} - \frac{3125 x ^{24}}{24} + C