integral of (x+2)^2*ln(5x)

解答

\int (x + 2)^{2} \cdot ln (5 x) d x

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

求解步骤

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

乘开

(x + 2)^{2} ln (5 x) : x^{2} ln (5 x) + 4 x ln (5 x) + 4 ln (5 x)

= \int x^{2} ln (5 x) + 4 x ln (5 x) + 4 ln (5 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 x^{2} ln (5 x) d x + \int 4 x ln (5 x) d x + \int 4 ln (5 x) d x

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

\int 4 x ln (5 x) d x = 2 x^{2} ln (5 x) - x^{2}

\int 4 ln (5 x) d x = 4 (x ln (5 x) - x)

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

解答补常数

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

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