integral from 1 to 2 of ((x-1)^3)/(x^2)

解答

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

3 ln (2) - 2

十进制

0.07944 \dots

求解步骤

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

乘开

\frac{( x - 1 ) ^{3}}{x ^{2}} : x - 3 + \frac{3}{x} - \frac{1}{x ^{2}}

= \int_{1}^{2} x - 3 + \frac{3}{x} - \frac{1}{x ^{2}} d x

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

\int_{1}^{2} x d x = \frac{3}{2}

\int_{1}^{2} 3 d x = 3

\int_{1}^{2} \frac{3}{x} d x = 3 ln (2)

\int_{1}^{2} \frac{1}{x ^{2}} d x = \frac{1}{2}

= \frac{3}{2} - 3 + 3 ln (2) - \frac{1}{2}

化简

\frac{3}{2} - 3 + 3 ln (2) - \frac{1}{2} : 3 ln (2) - 2

= 3 ln (2) - 2

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