integral of x^3(x^4-5)^7

解

\int x^{3} (x^{4} - 5)^{7} d x

\frac{x ^{32}}{32} - \frac{5 x ^{28}}{4} + \frac{175 x ^{24}}{8} - \frac{875 x ^{20}}{4} + \frac{21875 x ^{16}}{16} - \frac{21875 x ^{12}}{4} + \frac{109375 x ^{8}}{8} - \frac{78125 x ^{4}}{4} + C

解答ステップ

\int x^{3} (x^{4} - 5)^{7} d x

拡張

x^{3} (x^{4} - 5)^{7} : x^{31} - 35 x^{27} + 525 x^{23} - 4375 x^{19} + 21875 x^{15} - 65625 x^{11} + 109375 x^{7} - 78125 x^{3}

= \int x^{31} - 35 x^{27} + 525 x^{23} - 4375 x^{19} + 21875 x^{15} - 65625 x^{11} + 109375 x^{7} - 78125 x^{3} 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^{31} d x - \int 35 x^{27} d x + \int 525 x^{23} d x - \int 4375 x^{19} d x + \int 21875 x^{15} d x - \int 65625 x^{11} d x + \int 109375 x^{7} d x - \int 78125 x^{3} d x

\int x^{31} d x = \frac{x ^{32}}{32}

\int 35 x^{27} d x = \frac{5 x ^{28}}{4}

\int 525 x^{23} d x = \frac{175 x ^{24}}{8}

\int 4375 x^{19} d x = \frac{875 x ^{20}}{4}

\int 21875 x^{15} d x = \frac{21875 x ^{16}}{16}

\int 65625 x^{11} d x = \frac{21875 x ^{12}}{4}

\int 109375 x^{7} d x = \frac{109375 x ^{8}}{8}

\int 78125 x^{3} d x = \frac{78125 x ^{4}}{4}

= \frac{x ^{32}}{32} - \frac{5 x ^{28}}{4} + \frac{175 x ^{24}}{8} - \frac{875 x ^{20}}{4} + \frac{21875 x ^{16}}{16} - \frac{21875 x ^{12}}{4} + \frac{109375 x ^{8}}{8} - \frac{78125 x ^{4}}{4}

定数を解答に追加する

= \frac{x ^{32}}{32} - \frac{5 x ^{28}}{4} + \frac{175 x ^{24}}{8} - \frac{875 x ^{20}}{4} + \frac{21875 x ^{16}}{16} - \frac{21875 x ^{12}}{4} + \frac{109375 x ^{8}}{8} - \frac{78125 x ^{4}}{4} + C