integral from 1 to 5 of (x^2+5)/(6x-x^2)

解答

\int_{1}^{5} \frac{x ^{2} + 5}{6 x - x ^{2}} d x

- 4 + \frac{23}{3} ln (5)

十进制

8.33902 \dots

求解步骤

\int_{1}^{5} \frac{x ^{2} + 5}{6 x - x ^{2}} d x

分解

\frac{x ^{2} + 5}{6 x - x ^{2}}

= \int_{1}^{5} \frac{x ^{2} + 5}{- x ( x - 6 )} d x

提出常数:

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

= - \int_{1}^{5} \frac{x ^{2} + 5}{x ( x - 6 )} d x

乘开

\frac{x ^{2} + 5}{x ( x - 6 )} : \frac{x ^{2}}{x ( x - 6 )} + \frac{5}{x ( x - 6 )}

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

= - (\int_{1}^{5} \frac{x ^{2}}{x ( x - 6 )} d x + \int_{1}^{5} \frac{5}{x ( x - 6 )} d x)

\int_{1}^{5} \frac{x ^{2}}{x ( x - 6 )} d x = 4 - 6 ln (5)

\int_{1}^{5} \frac{5}{x ( x - 6 )} d x = - \frac{5}{3} ln (5)

= - (4 - 6 ln (5) - \frac{5}{3} ln (5))

化简

- (4 - 6 ln (5) - \frac{5}{3} ln (5)) : - 4 + \frac{23}{3} ln (5)

= - 4 + \frac{23}{3} ln (5)

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