integral from 0 to 2 of (2x-1)/((x+3)^2)

解答

\int_{0}^{2} \frac{2 x - 1}{( x + 3 ) ^{2}} d x

2 ln (5) - 2 ln (3) - \frac{14}{15}

十进制

0.08831 \dots

求解步骤

\int_{0}^{2} \frac{2 x - 1}{( x + 3 ) ^{2}} d x

乘开

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

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_{0}^{2} \frac{2 x}{( x + 3 ) ^{2}} d x - \int_{0}^{2} \frac{1}{( x + 3 ) ^{2}} d x

\int_{0}^{2} \frac{2 x}{( x + 3 ) ^{2}} d x = 2 (ln (5) - ln (3) - \frac{2}{5})

\int_{0}^{2} \frac{1}{( x + 3 ) ^{2}} d x = \frac{2}{15}

= 2 (ln (5) - ln (3) - \frac{2}{5}) - \frac{2}{15}

化简

2 (ln (5) - ln (3) - \frac{2}{5}) - \frac{2}{15} : 2 ln (5) - 2 ln (3) - \frac{14}{15}

= 2 ln (5) - 2 ln (3) - \frac{14}{15}

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