integral from 0 to 2 of 1/(1+x^4)

解答

\int_{0}^{2} \frac{1}{1 + x ^{4}} d x

\frac{2 ln ( 2 2 + 5 ) - 2 ln ( 5 - 2 2 ) + 4 arctan ( 2 2 + 1 ) + 4 arctan ( 2 2 - 1 )}{8 2}

十进制

1.07012 \dots

求解步骤

\int_{0}^{2} \frac{1}{1 + x ^{4}} d x

将

\frac{1}{1 + x ^{4}}

用部份分式展开:

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

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

\int_{0}^{2} \frac{x + 2}{2 2 ( x ^{2} + 2 x + 1 )} d x = \frac{1}{2} (\frac{1}{2 2} ln (22 + 5) + \frac{1}{2} (arctan (22 + 1) - \frac{π}{4}))

\int_{0}^{2} \frac{- x + 2}{2 2 ( x ^{2} - 2 x + 1 )} d x = \frac{1}{2} (- \frac{1}{2 2} ln (5 - 22) + \frac{1}{2} (arctan (22 - 1) + \frac{π}{4}))

= \frac{1}{2} (\frac{1}{2 2} ln (22 + 5) + \frac{1}{2} (arctan (22 + 1) - \frac{π}{4})) + \frac{1}{2} (- \frac{1}{2 2} ln (5 - 22) + \frac{1}{2} (arctan (22 - 1) + \frac{π}{4}))

化简

\frac{1}{2} (\frac{1}{2 2} ln (22 + 5) + \frac{1}{2} (arctan (22 + 1) - \frac{π}{4})) + \frac{1}{2} (- \frac{1}{2 2} ln (5 - 22) + \frac{1}{2} (arctan (22 - 1) + \frac{π}{4})) : \frac{2 ln ( 2 2 + 5 ) - 2 ln ( 5 - 2 2 ) + 4 arctan ( 2 2 + 1 ) + 4 arctan ( 2 2 - 1 )}{8 2}

= \frac{2 ln ( 2 2 + 5 ) - 2 ln ( 5 - 2 2 ) + 4 arctan ( 2 2 + 1 ) + 4 arctan ( 2 2 - 1 )}{8 2}

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