integral from 1/5 to 7 of 8xln(5x)

解答

\int_{\frac{1}{5}}^{7} 8 x ln (5 x) d x

8 (- \frac{306}{25} + \frac{49}{2} ln (35))

十进制

598.92822 \dots

求解步骤

\int_{\frac{1}{5}}^{7} 8 x ln (5 x) d x

提出常数:

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

= 8 \cdot \int_{\frac{1}{5}}^{7} x ln (5 x) d x

使用分布积分法

= 8 [\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x]_{\frac{1}{5}}^{7}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 8 [\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}]_{\frac{1}{5}}^{7}

带入上下限计算后得:

- \frac{306}{25} + \frac{49}{2} ln (35)

= 8 (- \frac{306}{25} + \frac{49}{2} ln (35))

y^{^{'}} + \frac{2}{x} y = 4 x

\frac{d}{d x} (ln (7 x^{2} + 5 y^{2}))

d er i v a t i v e 4 t^{\frac{- 1}{8}} + 5 ln (t) + ln (5)

t an g e n t f (x) = 3 x^{2} - 4 x + 1

\frac{d}{d x} (\frac{9 - sec ( x )}{tan ( x )})