integral from 1 to infinity of x/(1+x^4)

解答

\int_{1}^{\infty} \frac{x}{1 + x ^{4}} d x

\frac{π}{8}

十进制

0.39269 \dots

求解步骤

\int_{1}^{\infty} \frac{x}{1 + x ^{4}} d x

计算不定积分:

\int \frac{x}{1 + x ^{4}} d x = \frac{1}{2} arctan (x^{2}) + C

带入上下限计算后得:

\int_{1}^{\infty} \frac{x}{1 + x ^{4}} d x = \frac{π}{4} - \frac{π}{8}

= \frac{π}{4} - \frac{π}{8}

化简

= \frac{π}{8}

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