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

解答

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

- \frac{1}{3} ln (2) + \frac{π}{3 3}

十进制

0.37355 \dots

求解步骤

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

将

\frac{x}{x ^{3} + 1}

用部份分式展开:

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

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

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

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

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

\frac{d}{d x} (\frac{x - 6}{x ^{2} - 2 x + 12})

x \to 0 lim (- 2 sin (x))

t an g e n t y = 3 x^{3} - 2 x, (1, 1)

x \to 0 lim (ln (- x))

x \to \infty lim (arctan (\frac{x + 1}{x + 2}))