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

解答

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

2 ln (2) - ln (3) - \frac{1}{4}

十进制

0.03768 \dots

求解步骤

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

使用换元积分法

= \int_{3}^{4} \frac{u - 3}{u ^{2}} d u

乘开

\frac{u - 3}{u ^{2}} : \frac{1}{u} - \frac{3}{u ^{2}}

= \int_{3}^{4} \frac{1}{u} - \frac{3}{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_{3}^{4} \frac{1}{u} d u - \int_{3}^{4} \frac{3}{u ^{2}} d u

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

\int_{3}^{4} \frac{3}{u ^{2}} d u = \frac{1}{4}

= 2 ln (2) - ln (3) - \frac{1}{4}

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