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

解答

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

5 (ln (\frac{3}{2}) - \frac{5}{12})

十进制

- 0.05600 \dots

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

使用换元积分法

= 5 \cdot \int_{4}^{6} \frac{u - 5}{u ^{2}} d u

乘开

\frac{u - 5}{u ^{2}} : \frac{1}{u} - \frac{5}{u ^{2}}

= 5 \cdot \int_{4}^{6} \frac{1}{u} - \frac{5}{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

= 5 (\int_{4}^{6} \frac{1}{u} d u - \int_{4}^{6} \frac{5}{u ^{2}} d u)

\int_{4}^{6} \frac{1}{u} d u = ln (\frac{3}{2})

\int_{4}^{6} \frac{5}{u ^{2}} d u = \frac{5}{12}

= 5 (ln (\frac{3}{2}) - \frac{5}{12})

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