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受欢迎的微积分 >

integral of x^2*ln(2x-3)

解答

\int x^{2} \cdot ln (2 x - 3) d x

解答

\frac{1}{72} (3 ln (2 x - 3) (2 x - 3)^{3} - (2 x - 3)^{3} + 54 (\frac{ln ( 2 x - 3 ) ( 2 x - 3 ) ^{2}}{2} - \frac{( 2 x - 3 ) ^{2}}{4}) + 81 (2 x ln (2 x - 3) - 3 ln (2 x - 3) - 2 x + 3)) + C

求解步骤

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

使用换元积分法

= \int \frac{ln ( u ) ( u + 3 ) ^{2}}{8} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{8} \cdot \int ln (u) (u + 3)^{2} d u

乘开

ln (u) (u + 3)^{2} : u^{2} ln (u) + 6 u ln (u) + 9 ln (u)

= \frac{1}{8} \cdot \int u^{2} ln (u) + 6 u ln (u) + 9 ln (u) d u

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

= \frac{1}{8} (\int u^{2} ln (u) d u + \int 6 u ln (u) d u + \int 9 ln (u) d u)

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

\int 6 u ln (u) d u = 6 (\frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4})

\int 9 ln (u) d u = 9 (u ln (u) - u)

= \frac{1}{8} (\frac{1}{3} u^{3} ln (u) - \frac{u ^{3}}{9} + 6 (\frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4}) + 9 (u ln (u) - u))

u = 2 x - 3

代回

= \frac{1}{8} (\frac{1}{3} (2 x - 3)^{3} ln (2 x - 3) - \frac{( 2 x - 3 ) ^{3}}{9} + 6 (\frac{1}{2} (2 x - 3)^{2} ln (2 x - 3) - \frac{( 2 x - 3 ) ^{2}}{4}) + 9 ((2 x - 3) ln (2 x - 3) - (2 x - 3)))

化简

\frac{1}{8} (\frac{1}{3} (2 x - 3)^{3} ln (2 x - 3) - \frac{( 2 x - 3 ) ^{3}}{9} + 6 (\frac{1}{2} (2 x - 3)^{2} ln (2 x - 3) - \frac{( 2 x - 3 ) ^{2}}{4}) + 9 ((2 x - 3) ln (2 x - 3) - (2 x - 3))) : \frac{1}{72} (3 ln (2 x - 3) (2 x - 3)^{3} - (2 x - 3)^{3} + 54 (\frac{ln ( 2 x - 3 ) ( 2 x - 3 ) ^{2}}{2} - \frac{( 2 x - 3 ) ^{2}}{4}) + 81 (2 x ln (2 x - 3) - 3 ln (2 x - 3) - 2 x + 3))

= \frac{1}{72} (3 ln (2 x - 3) (2 x - 3)^{3} - (2 x - 3)^{3} + 54 (\frac{ln ( 2 x - 3 ) ( 2 x - 3 ) ^{2}}{2} - \frac{( 2 x - 3 ) ^{2}}{4}) + 81 (2 x ln (2 x - 3) - 3 ln (2 x - 3) - 2 x + 3))

解答补常数

= \frac{1}{72} (3 ln (2 x - 3) (2 x - 3)^{3} - (2 x - 3)^{3} + 54 (\frac{ln ( 2 x - 3 ) ( 2 x - 3 ) ^{2}}{2} - \frac{( 2 x - 3 ) ^{2}}{4}) + 81 (2 x ln (2 x - 3) - 3 ln (2 x - 3) - 2 x + 3)) + C

作图

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