integral of (3/8 x-10)^5

解

\int (\frac{3}{8} x - 10)^{5} d x

\frac{81 x ^{6}}{65536} - \frac{405 x ^{5}}{2048} + \frac{3375 x ^{4}}{256} - \frac{1875 x ^{3}}{4} + 9375 x^{2} - 100000 x + C

解答ステップ

\int (\frac{3}{8} x - 10)^{5} d x

拡張

(\frac{3}{8} x - 10)^{5} : \frac{243 x ^{5}}{32768} - \frac{2025 x ^{4}}{2048} + \frac{3375 x ^{3}}{64} - \frac{5625 x ^{2}}{4} + 18750 x - 100000

= \int \frac{243 x ^{5}}{32768} - \frac{2025 x ^{4}}{2048} + \frac{3375 x ^{3}}{64} - \frac{5625 x ^{2}}{4} + 18750 x - 100000 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 \frac{243 x ^{5}}{32768} d x - \int \frac{2025 x ^{4}}{2048} d x + \int \frac{3375 x ^{3}}{64} d x - \int \frac{5625 x ^{2}}{4} d x + \int 18750 x d x - \int 100000 d x

\int \frac{243 x ^{5}}{32768} d x = \frac{81 x ^{6}}{65536}

\int \frac{2025 x ^{4}}{2048} d x = \frac{405 x ^{5}}{2048}

\int \frac{3375 x ^{3}}{64} d x = \frac{3375 x ^{4}}{256}

\int \frac{5625 x ^{2}}{4} d x = \frac{1875 x ^{3}}{4}

\int 18750 x d x = 9375 x^{2}

\int 100000 d x = 100000 x

= \frac{81 x ^{6}}{65536} - \frac{405 x ^{5}}{2048} + \frac{3375 x ^{4}}{256} - \frac{1875 x ^{3}}{4} + 9375 x^{2} - 100000 x

定数を解答に追加する

= \frac{81 x ^{6}}{65536} - \frac{405 x ^{5}}{2048} + \frac{3375 x ^{4}}{256} - \frac{1875 x ^{3}}{4} + 9375 x^{2} - 100000 x + C