integral from 1 to 5 of 1+1/u+1/(u^2)

解答

\int_{1}^{5} 1 + \frac{1}{u} + \frac{1}{u ^{2}} d u

ln (5) + \frac{24}{5}

十进制

6.40943 \dots

求解步骤

\int_{1}^{5} 1 + \frac{1}{u} + \frac{1}{u ^{2}} d u

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} 1 d u + \int_{1}^{5} \frac{1}{u} d u + \int_{1}^{5} \frac{1}{u ^{2}} d u

\int_{1}^{5} 1 d u = 4

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

\int_{1}^{5} \frac{1}{u ^{2}} d u = \frac{4}{5}

= 4 + ln (5) + \frac{4}{5}

合并分式

4 + \frac{4}{5} : \frac{24}{5}

= ln (5) + \frac{24}{5}

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