integral of (2x^3+5x^2-3x+1)ln(x)

解答

\int (2 x^{3} + 5 x^{2} - 3 x + 1) ln (x) d x

2 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 5 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) - 3 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x + C

求解步骤

\int (2 x^{3} + 5 x^{2} - 3 x + 1) ln (x) d x

乘开

(2 x^{3} + 5 x^{2} - 3 x + 1) ln (x) : 2 x^{3} ln (x) + 5 x^{2} ln (x) - 3 x ln (x) + ln (x)

= \int 2 x^{3} ln (x) + 5 x^{2} ln (x) - 3 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 2 x^{3} ln (x) d x + \int 5 x^{2} ln (x) d x - \int 3 x ln (x) d x + \int ln (x) d x

\int 2 x^{3} ln (x) d x = 2 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

\int 5 x^{2} ln (x) d x = 5 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

\int 3 x ln (x) d x = 3 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4})

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

= 2 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 5 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) - 3 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x

解答补常数

= 2 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + 5 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) - 3 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) + x ln (x) - x + C

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