integral from 2 to 3 of (ln(2x^5))/(x^2)

解答

\int_{2}^{3} \frac{ln ( 2 x ^{5} )}{x ^{2}} d x

\frac{5 + ln ( \frac{65536}{59049} )}{6}

十进制

0.85070 \dots

求解步骤

\int_{2}^{3} \frac{ln ( 2 x ^{5} )}{x ^{2}} d x

使用分布积分法

= [- \frac{ln ( 2 x ^{5} )}{x} - \int - \frac{5}{x ^{2}} d x]_{2}^{3}

\int - \frac{5}{x ^{2}} d x = \frac{5}{x}

= [- \frac{ln ( 2 x ^{5} )}{x} - \frac{5}{x}]_{2}^{3}

化简

- \frac{ln ( 2 x ^{5} )}{x} - \frac{5}{x} : \frac{- ln ( 2 ) - 5 ln ( x ) - 5}{x}

= [\frac{- ln ( 2 ) - 5 ln ( x ) - 5}{x}]_{2}^{3}

带入上下限计算后得:

\frac{5 + 18 ln ( 2 ) - 2 ln ( 486 )}{6}

= \frac{5 + 18 ln ( 2 ) - 2 ln ( 486 )}{6}

化简

= \frac{5 + ln ( \frac{65536}{59049} )}{6}

\frac{\partial}{\partial x} (x^{a} y^{1 - a})

\frac{\partial}{\partial x} (sin (x + 2 y))

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

x \to - 2 - lim (\frac{3 x - 6}{x + 2})

\frac{d}{d x} (\frac{π}{4} x)