integral of (6x^2+1)ln(5x^3)

解答

\int (6 x^{2} + 1) ln (5 x^{3}) d x

2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x + C

求解步骤

\int (6 x^{2} + 1) ln (5 x^{3}) d x

使用分布积分法

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

\int 3 (2 x^{2} + 1) d x = 2 x^{3} + 3 x

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

化简

ln (5 x^{3}) (2 x^{3} + x) - (2 x^{3} + 3 x) : 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x

= 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x

解答补常数

= 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x + C

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