integral of xln(5+x)

解答

\int x ln (5 + x) d x

ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

求解步骤

\int x ln (5 + x) d x

使用换元积分法

= \int (u - 5) ln (u) d u

使用分布积分法

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u

\int \frac{1}{2} (u - 10) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

u = 5 + x

代回

= ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x))

化简

ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x)) : ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

解答补常数

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

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