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

解答

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

\frac{8}{25}

十进制

0.32

求解步骤

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

将

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

用部份分式展开:

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

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

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

\int_{1}^{3} \frac{1}{2 ( 2 x - 1 ) ^{3}} d x = \frac{3}{25}

= \frac{1}{5} + \frac{3}{25}

化简

\frac{1}{5} + \frac{3}{25} : \frac{8}{25}

= \frac{8}{25}

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