integral of ln(5x+8)

解答

\int ln (5 x + 8) d x

x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5} + C

求解步骤

\int ln (5 x + 8) d x

使用换元积分法

= \int ln (u) \frac{1}{5} d u

提出常数:

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

= \frac{1}{5} \cdot \int ln (u) d u

使用分布积分法

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

\int 1 d u = u

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

u = 5 x + 8

代回

= \frac{1}{5} ((5 x + 8) ln (5 x + 8) - (5 x + 8))

化简

\frac{1}{5} ((5 x + 8) ln (5 x + 8) - (5 x + 8)) : x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5}

= x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5}

解答补常数

= x ln (5 x + 8) + \frac{8}{5} ln (5 x + 8) - x - \frac{8}{5} + C

\frac{d}{d x} (\frac{1 - 3 x ^{2}}{1 - 2 x})

\frac{d}{d x} (9 x + 5)

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