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受欢迎的微积分 >

integral from 1 to 4 of (x^2+8)/(5x-x^2)

解答

\int_{1}^{4} \frac{x ^{2} + 8}{5 x - x ^{2}} d x

解答

- 3 + \frac{82}{5} ln (2)

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十进制

8.36761 \dots

求解步骤

\int_{1}^{4} \frac{x ^{2} + 8}{5 x - x ^{2}} d x

分解

\frac{x ^{2} + 8}{5 x - x ^{2}}

= \int_{1}^{4} \frac{x ^{2} + 8}{- x ( x - 5 )} d x

提出常数:

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

= - \int_{1}^{4} \frac{x ^{2} + 8}{x ( x - 5 )} d x

乘开

\frac{x ^{2} + 8}{x ( x - 5 )} : \frac{x ^{2}}{x ( x - 5 )} + \frac{8}{x ( x - 5 )}

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}^{4} \frac{x ^{2}}{x ( x - 5 )} d x + \int_{1}^{4} \frac{8}{x ( x - 5 )} d x)

\int_{1}^{4} \frac{x ^{2}}{x ( x - 5 )} d x = 3 - 10 ln (2)

\int_{1}^{4} \frac{8}{x ( x - 5 )} d x = - \frac{32}{5} ln (2)

= - (3 - 10 ln (2) - \frac{32}{5} ln (2))

化简

- (3 - 10 ln (2) - \frac{32}{5} ln (2)) : - 3 + \frac{82}{5} ln (2)

= - 3 + \frac{82}{5} ln (2)

作图

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