integral of (4x^2-5x+4)^4(8x-5)

解

\int (4 x^{2} - 5 x + 4)^{4} (8 x - 5) d x

\frac{1024 x ^{10}}{5} - 1280 x^{9} + 4224 x^{8} - 9120 x^{7} + 14148 x^{6} - 16305 x^{5} + 14148 x^{4} - 9120 x^{3} + 4224 x^{2} - 1280 x + C

解答ステップ

\int (4 x^{2} - 5 x + 4)^{4} (8 x - 5) d x

拡張

(4 x^{2} - 5 x + 4)^{4} (8 x - 5) : 2048 x^{9} - 11520 x^{8} + 33792 x^{7} - 63840 x^{6} + 84888 x^{5} - 81525 x^{4} + 56592 x^{3} - 27360 x^{2} + 8448 x - 1280

= \int 2048 x^{9} - 11520 x^{8} + 33792 x^{7} - 63840 x^{6} + 84888 x^{5} - 81525 x^{4} + 56592 x^{3} - 27360 x^{2} + 8448 x - 1280 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 11520 x^{8} d x + \int 33792 x^{7} d x - \int 63840 x^{6} d x + \int 84888 x^{5} d x - \int 81525 x^{4} d x + \int 56592 x^{3} d x - \int 27360 x^{2} d x + \int 8448 x d x - \int 1280 d x

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

\int 11520 x^{8} d x = 1280 x^{9}

\int 33792 x^{7} d x = 4224 x^{8}

\int 63840 x^{6} d x = 9120 x^{7}

\int 84888 x^{5} d x = 14148 x^{6}

\int 81525 x^{4} d x = 16305 x^{5}

\int 56592 x^{3} d x = 14148 x^{4}

\int 27360 x^{2} d x = 9120 x^{3}

\int 8448 x d x = 4224 x^{2}

\int 1280 d x = 1280 x

= \frac{1024 x ^{10}}{5} - 1280 x^{9} + 4224 x^{8} - 9120 x^{7} + 14148 x^{6} - 16305 x^{5} + 14148 x^{4} - 9120 x^{3} + 4224 x^{2} - 1280 x

定数を解答に追加する

= \frac{1024 x ^{10}}{5} - 1280 x^{9} + 4224 x^{8} - 9120 x^{7} + 14148 x^{6} - 16305 x^{5} + 14148 x^{4} - 9120 x^{3} + 4224 x^{2} - 1280 x + C