integral of (2x+1)^7ln(x)

解答

\int (2 x + 1)^{7} ln (x) d x

128 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) + 448 (\frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49}) + 672 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36}) + 560 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}) + 280 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 84 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + 14 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x + C

求解步骤

\int (2 x + 1)^{7} ln (x) d x

乘开

(2 x + 1)^{7} ln (x) : 128 x^{7} ln (x) + 448 x^{6} ln (x) + 672 x^{5} ln (x) + 560 x^{4} ln (x) + 280 x^{3} ln (x) + 84 x^{2} ln (x) + 14 x ln (x) + ln (x)

= \int 128 x^{7} ln (x) + 448 x^{6} ln (x) + 672 x^{5} ln (x) + 560 x^{4} ln (x) + 280 x^{3} ln (x) + 84 x^{2} ln (x) + 14 x ln (x) + ln (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 128 x^{7} ln (x) d x + \int 448 x^{6} ln (x) d x + \int 672 x^{5} ln (x) d x + \int 560 x^{4} ln (x) d x + \int 280 x^{3} ln (x) d x + \int 84 x^{2} ln (x) d x + \int 14 x ln (x) d x + \int ln (x) d x

\int 128 x^{7} ln (x) d x = 128 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64})

\int 448 x^{6} ln (x) d x = 448 (\frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49})

\int 672 x^{5} ln (x) d x = 672 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36})

\int 560 x^{4} ln (x) d x = 560 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25})

\int 280 x^{3} ln (x) d x = 280 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

\int 84 x^{2} ln (x) d x = 84 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

\int 14 x ln (x) d x = 14 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4})

\int ln (x) d x = x ln (x) - x

= 128 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) + 448 (\frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49}) + 672 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36}) + 560 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}) + 280 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 84 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + 14 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x

解答补常数

= 128 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) + 448 (\frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49}) + 672 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36}) + 560 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}) + 280 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 84 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + 14 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x + C

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