integral from 5 to x of 1/(1+t^4)

解答

\int_{5}^{x} \frac{1}{1 + t ^{4}} d t

\frac{1}{2} (\frac{1}{2 2} (ln 2 x^{2} + 22 x + 2 - ln (102 + 52)) + \frac{1}{2} (arctan (2 x + 1) - arctan (52 + 1))) + \frac{1}{2} (- \frac{1}{2 2} (ln 2 x^{2} - 22 x + 2 - ln (52 - 102)) + \frac{1}{2} (arctan (2 x - 1) - arctan (52 - 1)))

求解步骤

\int_{5}^{x} \frac{1}{1 + t ^{4}} d t

将

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

用部份分式展开:

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

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

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

\int_{5}^{x} \frac{t + 2}{2 2 ( t ^{2} + 2 t + 1 )} d t = \frac{1}{2} (\frac{1}{2 2} (ln 2 x^{2} + 22 x + 2 - ln (102 + 52)) + \frac{1}{2} (arctan (2 x + 1) - arctan (52 + 1)))

\int_{5}^{x} \frac{- t + 2}{2 2 ( t ^{2} - 2 t + 1 )} d t = \frac{1}{2} (- \frac{1}{2 2} (ln 2 x^{2} - 22 x + 2 - ln (52 - 102)) + \frac{1}{2} (arctan (2 x - 1) - arctan (52 - 1)))

= \frac{1}{2} (\frac{1}{2 2} (ln 2 x^{2} + 22 x + 2 - ln (102 + 52)) + \frac{1}{2} (arctan (2 x + 1) - arctan (52 + 1))) + \frac{1}{2} (- \frac{1}{2 2} (ln 2 x^{2} - 22 x + 2 - ln (52 - 102)) + \frac{1}{2} (arctan (2 x - 1) - arctan (52 - 1)))

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