integral of (x+3)ln(2x)

解答

\int (x + 3) ln (2 x) d x

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

求解步骤

\int (x + 3) ln (2 x) d x

乘开

(x + 3) ln (2 x) : x ln (2 x) + 3 ln (2 x)

= \int x ln (2 x) + 3 ln (2 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 ln (2 x) d x + \int 3 ln (2 x) d x

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

\int 3 ln (2 x) d x = 3 (x ln (2 x) - x)

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

解答补常数

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

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