integral of (4x^2-3x)^4(8x-3)

解

\int (4 x^{2} - 3 x)^{4} (8 x - 3) d x

\frac{1024 x ^{10}}{5} - 768 x^{9} + 1152 x^{8} - 864 x^{7} + 324 x^{6} - \frac{243 x ^{5}}{5} + C

解答ステップ

\int (4 x^{2} - 3 x)^{4} (8 x - 3) d x

拡張

(4 x^{2} - 3 x)^{4} (8 x - 3) : 2048 x^{9} - 6912 x^{8} + 9216 x^{7} - 6048 x^{6} + 1944 x^{5} - 243 x^{4}

= \int 2048 x^{9} - 6912 x^{8} + 9216 x^{7} - 6048 x^{6} + 1944 x^{5} - 243 x^{4} 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 2048 x^{9} d x - \int 6912 x^{8} d x + \int 9216 x^{7} d x - \int 6048 x^{6} d x + \int 1944 x^{5} d x - \int 243 x^{4} d x

\int 2048 x^{9} d x = \frac{1024 x ^{10}}{5}

\int 6912 x^{8} d x = 768 x^{9}

\int 9216 x^{7} d x = 1152 x^{8}

\int 6048 x^{6} d x = 864 x^{7}

\int 1944 x^{5} d x = 324 x^{6}

\int 243 x^{4} d x = \frac{243 x ^{5}}{5}

= \frac{1024 x ^{10}}{5} - 768 x^{9} + 1152 x^{8} - 864 x^{7} + 324 x^{6} - \frac{243 x ^{5}}{5}

定数を解答に追加する

= \frac{1024 x ^{10}}{5} - 768 x^{9} + 1152 x^{8} - 864 x^{7} + 324 x^{6} - \frac{243 x ^{5}}{5} + C